{"id":279,"date":"2021-10-30T21:31:37","date_gmt":"2021-10-30T13:31:37","guid":{"rendered":"https:\/\/www.lazybirds.top\/?p=279"},"modified":"2021-10-30T21:31:37","modified_gmt":"2021-10-30T13:31:37","slug":"%e7%bb%84%e5%90%88%e6%95%b0%e5%ad%a6-%e6%9c%9f%e4%b8%ad%e5%a4%8d%e4%b9%a0","status":"publish","type":"post","link":"https:\/\/www.lazybirds.top\/?p=279","title":{"rendered":"\u7ec4\u5408\u6570\u5b66 &#8211; \u671f\u4e2d\u590d\u4e60"},"content":{"rendered":"\n<p>\u7531VScode\u63d2\u4ef6Markdown Preview Enhanced\u751f\u6210html\uff0c\u5c06\u4f9d\u8d56\u7684katex\u6587\u4ef6\u62f7\u8d1d\u5230\u7f51\u7ad9\u5373\u53ef<\/p>\n\n\n\n<!--more-->\n\n\n\n\n      <meta charset=\"utf-8\">\n      <meta name=\"viewport\" content=\"width=device-width, initial-scale=1.0\">\n      \n      <link rel=\"stylesheet\" href=\"https:\/\/www.lazybirds.top\/katex\/katex.min.css\">\n      \n      \n      \n      \n      \n      \n      \n      \n      \n      <style>\n      \/**\n * prism.js Github theme 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reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">&#x3C9;<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.664392em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><munder><mo>&#x2211;<\/mo><mrow><mi>k<\/mi><mo>&#x2265;<\/mo><mn>0<\/mn><\/mrow><\/munder><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>n<\/mi><mi>k<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mi>k<\/mi><msup><mi>x<\/mi><mi>k<\/mi><\/msup><mo>=<\/mo><mi>n<\/mi><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>x<\/mi><msup><mo stretchy=\"false\">)<\/mo><mrow><mi>n<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\sum_{k\\ge 0}\\binom{n}{k}kx^{k}=nx(1+x)^{n-1}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.847292em;vertical-align:-1.397292em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.050005em;\"><span style=\"top:-1.8478869999999998em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mrel mtight\">&#x2265;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span style=\"top:-3.0500049999999996em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">&#x2211;<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.397292em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.1075599999999999em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8991079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1.1141079999999999em;vertical-align:-0.25em;\"><\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.864108em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">&#x2212;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><munder><mo>&#x2211;<\/mo><mrow><mi>k<\/mi><mo>&#x2265;<\/mo><mn>0<\/mn><\/mrow><\/munder><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>n<\/mi><mi>k<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mfrac><msup><mi>x<\/mi><mi>k<\/mi><\/msup><mi>k<\/mi><\/mfrac><mo>=<\/mo><msubsup><mo>&#x222B;<\/mo><mn>0<\/mn><mi>x<\/mi><\/msubsup><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>x<\/mi><msup><mo stretchy=\"false\">)<\/mo><mi>n<\/mi><\/msup><\/mrow><mi>x<\/mi><\/mfrac><mtext>&#xA0;<\/mtext><mi mathvariant=\"bold\">d<\/mi><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\sum_{k\\ge 0}\\binom{n}{k}\\frac{x^k}{k}=\\int_0^x\\frac{(1+x)^n}{x}~\\mathbf{d}x<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.9234em;vertical-align:-1.397292em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.050005em;\"><span style=\"top:-1.8478869999999998em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mrel mtight\">&#x2265;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span style=\"top:-3.0500049999999996em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">&#x2211;<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.397292em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.1075599999999999em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.526108em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.849108em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.33895em;vertical-align:-0.9119499999999999em;\"><\/span><span class=\"mop\"><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;\">&#x222B;<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.414292em;\"><span style=\"top:-1.7880500000000001em;margin-left:-0.44445em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><span style=\"top:-3.8129000000000004em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9119499999999999em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.427em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.664392em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace nobreak\">&#xA0;<\/span><span class=\"mord mathbf\">d<\/span><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><\/span><\/p>\n<h2 class=\"mume-header\" id=\"vandermonde-%E6%81%92%E7%AD%89%E5%BC%8F\">Vandermonde &#x6052;&#x7B49;&#x5F0F;<\/h2>\n\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><munder><mo>&#x2211;<\/mo><mrow><mn>0<\/mn><mo>&#x2264;<\/mo><mi>i<\/mi><mo>&#x2264;<\/mo><mi>l<\/mi><\/mrow><\/munder><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>n<\/mi><mi>i<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>m<\/mi><mrow><mi>l<\/mi><mo>&#x2212;<\/mo><mi>i<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mi>n<\/mi><mo>+<\/mo><mi>m<\/mi><\/mrow><mi>l<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\sum_{0\\le i\\le l}\\binom{n}{i}\\binom{m}{l-i}=\\binom{n+m}{l}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.847292em;vertical-align:-1.397292em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.050005em;\"><span style=\"top:-1.8478869999999998em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">0<\/span><span class=\"mrel mtight\">&#x2264;<\/span><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#x2264;<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.01968em;\">l<\/span><\/span><\/span><\/span><span style=\"top:-3.0500049999999996em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">&#x2211;<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.397292em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.1075599999999999em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.10756em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.01968em;\">l<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\">i<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7693300000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.40003em;vertical-align:-0.95003em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2603300000000002em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.01968em;\">l<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>k<\/mi><mo>=<\/mo><mi>n<\/mi><mtext>&#x65F6;&#xFF0C;<\/mtext><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><mi>n<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><munder><mo>&#x2211;<\/mo><mrow><mn>0<\/mn><mo>&#x2264;<\/mo><mi>i<\/mi><mo>&#x2264;<\/mo><mi>n<\/mi><\/mrow><\/munder><msup><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>n<\/mi><mi>i<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">k=n\\text{&#x65F6;&#xFF0C;}\\binom{2n}{n}=\\sum_{0\\le i\\le n}\\binom{n}{i}^2<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69444em;vertical-align:0em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.40003em;vertical-align:-0.95003em;\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord text\"><span class=\"mord cjk_fallback\">&#x65F6;&#xFF0C;<\/span><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span style=\"top:-3.6769999999999996em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:3.0268560000000004em;vertical-align:-1.372848em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.050005em;\"><span style=\"top:-1.8723309999999997em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">0<\/span><span class=\"mrel mtight\">&#x2264;<\/span><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#x2264;<\/span><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span style=\"top:-3.0500049999999996em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">&#x2211;<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.372848em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.1075599999999999em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.6540080000000001em;\"><span style=\"top:-3.9029000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h2 class=\"mume-header\" id=\"%E6%9D%A8%E8%BE%89%E4%B8%89%E8%A7%92\">&#x6768;&#x8F89;&#x4E09;&#x89D2;<\/h2>\n\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><mi>k<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>n<\/mi><mi>k<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>+<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>n<\/mi><mrow><mi>k<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\binom{n+1}{k}=\\binom{n}{k}+\\binom{n}{k-1}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.40003em;vertical-align:-0.95003em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.40003em;vertical-align:-0.95003em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.1075599999999999em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.40003em;vertical-align:-0.95003em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.10756em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7693300000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h2 class=\"mume-header\" id=\"%E6%9C%B1%E4%B8%96%E6%9D%B0%E6%81%92%E7%AD%89%E5%BC%8F\">&#x6731;&#x4E16;&#x6770;&#x6052;&#x7B49;&#x5F0F;<\/h2>\n\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><munder><mo>&#x2211;<\/mo><mrow><mn>0<\/mn><mo>&#x2264;<\/mo><mi>l<\/mi><mo>&#x2264;<\/mo><mi>n<\/mi><\/mrow><\/munder><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>l<\/mi><mi>k<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>k<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\sum_{0\\le l\\le n}\\binom{l}{k}=\\binom{n+1}{k+1}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.847292em;vertical-align:-1.397292em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.050005em;\"><span style=\"top:-1.8478869999999998em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">0<\/span><span class=\"mrel mtight\">&#x2264;<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.01968em;\">l<\/span><span class=\"mrel mtight\">&#x2264;<\/span><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span style=\"top:-3.0500049999999996em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">&#x2211;<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.397292em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.3714399999999998em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.01968em;\">l<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.40003em;vertical-align:-0.95003em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.3214400000000002em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><span style=\"top:-3.6770000000000005em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7693300000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>&#x5E94;&#x7528;<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.2500em\" columnalign=\"right left\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><munder><mo>&#x2211;<\/mo><mrow><mn>0<\/mn><mo>&#x2264;<\/mo><mi>m<\/mi><mo>&#x2264;<\/mo><mi>n<\/mi><\/mrow><\/munder><msup><mi>m<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mo>&#x2211;<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>m<\/mi><mn>1<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>+<\/mo><mn>2<\/mn><mo>&#x2211;<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>m<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>+<\/mo><mn>2<\/mn><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mi>n<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow><mn>3<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{aligned}\\sum_{0\\le m\\le n}m^2&amp;=\\sum\\binom{m}{1}+2\\sum\\binom{m}{2}\\\\\n&amp;=\\binom{n+1}{2}+2\\binom{n+2}{3}\\end{aligned}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:5.812322em;vertical-align:-2.656161em;\"><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-r\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:3.156161em;\"><span style=\"top:-5.156161em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.050005em;\"><span style=\"top:-1.8828869999999998em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">0<\/span><span class=\"mrel mtight\">&#x2264;<\/span><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mrel mtight\">&#x2264;<\/span><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span style=\"top:-3.0500049999999996em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">&#x2211;<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.362292em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">m<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-2.0438690000000004em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.656161em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:3.156161em;\"><span style=\"top:-5.156161em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mop op-symbol large-op\" style=\"position:relative;top:-0.000004999999999977245em;\">&#x2211;<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.1075599999999999em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mop op-symbol large-op\" style=\"position:relative;top:-0.000004999999999977245em;\">&#x2211;<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.1075599999999999em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><\/span><\/span><span style=\"top:-2.0438690000000004em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">3<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.656161em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.2500em\" columnalign=\"right left\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><munder><mo>&#x2211;<\/mo><mrow><mn>0<\/mn><mo>&#x2264;<\/mo><mi>m<\/mi><mo>&#x2264;<\/mo><mi>n<\/mi><\/mrow><\/munder><msup><mi>m<\/mi><mn>3<\/mn><\/msup><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mn>6<\/mn><mo>&#x2211;<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>m<\/mi><mn>3<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>+<\/mo><mn>3<\/mn><mo>&#x2211;<\/mo><msup><mi>m<\/mi><mn>2<\/mn><\/msup><mo>&#x2212;<\/mo><mn>2<\/mn><mo>&#x2211;<\/mo><mi>m<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mn>6<\/mn><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><mn>4<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>+<\/mo><mn>6<\/mn><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><mn>3<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>+<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{aligned}\\sum_{0\\le m\\le n}m^3&amp;=6\\sum\\binom{m}{3}+3\\sum m^2-2\\sum m\\\\\n&amp;=6\\binom{n+1}{4}+6\\binom{n+1}{3}+\\binom{n+1}{2}\\end{aligned}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:5.812322em;vertical-align:-2.656161em;\"><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-r\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:3.156161em;\"><span style=\"top:-5.156161em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.050005em;\"><span style=\"top:-1.8828869999999998em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">0<\/span><span class=\"mrel mtight\">&#x2264;<\/span><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mrel mtight\">&#x2264;<\/span><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span style=\"top:-3.0500049999999996em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">&#x2211;<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.362292em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">m<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-2.0438690000000004em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.656161em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:3.156161em;\"><span style=\"top:-5.156161em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">6<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mop op-symbol large-op\" style=\"position:relative;top:-0.000004999999999977245em;\">&#x2211;<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.1075599999999999em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">3<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">3<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mop op-symbol large-op\" style=\"position:relative;top:-0.000004999999999977245em;\">&#x2211;<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">m<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mop op-symbol large-op\" style=\"position:relative;top:-0.000004999999999977245em;\">&#x2211;<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><\/span><\/span><span style=\"top:-2.0438690000000004em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">6<\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">4<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">6<\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">3<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.656161em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h2 class=\"mume-header\" id=\"%E5%B9%BF%E4%B9%89%E7%BB%84%E5%90%88%E6%95%B0\">&#x5E7F;&#x4E49;&#x7EC4;&#x5408;&#x6570;<\/h2>\n\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mi>p<\/mi><mo>+<\/mo><mi>q<\/mi><mo>+<\/mo><mi>r<\/mi><\/mrow><mrow><mi>p<\/mi><mo separator=\"true\">,<\/mo><mi>q<\/mi><mo separator=\"true\">,<\/mo><mi>r<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>:<\/mo><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mi>p<\/mi><mo>+<\/mo><mi>q<\/mi><mo>+<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">!<\/mo><\/mrow><mrow><mi>p<\/mi><mo stretchy=\"false\">!<\/mo><mo>&#x22C5;<\/mo><mi>q<\/mi><mo stretchy=\"false\">!<\/mo><mo>&#x22C5;<\/mo><mi>r<\/mi><mo stretchy=\"false\">!<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\binom{p+q+r}{p,q,r}:=\\frac{(p+q+r)!}{p!\\cdot q!\\cdot r!}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.40003em;vertical-align:-0.95003em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2603300000000002em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">q<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r<\/span><\/span><\/span><span style=\"top:-3.6770000000000005em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">q<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8804400000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">:=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.30744em;vertical-align:-0.8804400000000001em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.427em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><span class=\"mclose\">!<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x22C5;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">q<\/span><span class=\"mclose\">!<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x22C5;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r<\/span><span class=\"mclose\">!<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">p<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">q<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r<\/span><span class=\"mclose\">)!<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8804400000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><mo>+<\/mo><mo>&#x22EF;<\/mo><mo>+<\/mo><msub><mi>a<\/mi><mi>m<\/mi><\/msub><msup><mo stretchy=\"false\">)<\/mo><mi>n<\/mi><\/msup><mo>=<\/mo><munder><mo>&#x2211;<\/mo><mtable rowspacing=\"0.2500em\" columnalign=\"center\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msub><mi>i<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>i<\/mi><mn>2<\/mn><\/msub><mo>+<\/mo><mo>&#x22EF;<\/mo><mo>+<\/mo><msub><mi>i<\/mi><mi>m<\/mi><\/msub><mo>=<\/mo><mi>n<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msub><mi>i<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>i<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><mo>&#x22EF;<\/mo><mtext>&#x2009;<\/mtext><mo separator=\"true\">,<\/mo><msub><mi>i<\/mi><mi>m<\/mi><\/msub><mo>&#x2265;<\/mo><mn>0<\/mn><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/munder><msubsup><mi>a<\/mi><mn>1<\/mn><msub><mi>i<\/mi><mn>1<\/mn><\/msub><\/msubsup><msubsup><mi>a<\/mi><mn>2<\/mn><msub><mi>i<\/mi><mn>2<\/mn><\/msub><\/msubsup><mo>&#x22EF;<\/mo><msubsup><mi>a<\/mi><mi>m<\/mi><msub><mi>i<\/mi><mi>m<\/mi><\/msub><\/msubsup><mfrac><mrow><mi>n<\/mi><mo stretchy=\"false\">!<\/mo><\/mrow><mrow><msub><mi>i<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">!<\/mo><msub><mi>i<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">!<\/mo><mo>&#x22EF;<\/mo><msub><mi>i<\/mi><mi>m<\/mi><\/msub><mo stretchy=\"false\">!<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">(a_1+a_2+\\cdots+a_m)^n=\\sum_{\\begin{gathered}i_1+i_2+\\cdots+i_m=n\\\\i_1,i_2,\\cdots,i_m\\ge0\\end{gathered}}a_1^{i_1}a_2^{i_2}\\cdots a_m^{i_m}\\frac{n!}{i_1!i_2!\\cdots i_m!}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.73333em;vertical-align:-0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.66666em;vertical-align:-0.08333em;\"><\/span><span class=\"minner\">&#x22EF;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7143919999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:4.430485em;vertical-align:-3.0590450000000002em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.050005em;\"><span style=\"top:-1.283995em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.29652em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.8521714285714288em;\"><span style=\"top:-3.91em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord sizing reset-size3 size6\"><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"minner\">&#x22EF;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span style=\"top:-2.3078285714285713em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord sizing reset-size3 size6\"><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\">&#x22EF;<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">&#x2265;<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.3521714285714288em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3.296525em;\"><span class=\"pstrut\" style=\"height:3.29652em;\"><\/span><span><span class=\"mop op-symbol large-op\">&#x2211;<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:3.0590450000000002em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9065639999999999em;\"><span style=\"top:-2.433692em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span style=\"top:-3.1449em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31731428571428577em;\"><span style=\"top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.26630799999999993em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9065639999999999em;\"><span style=\"top:-2.433692em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span style=\"top:-3.1449em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31731428571428577em;\"><span style=\"top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.26630799999999993em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\">&#x22EF;<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8746639999999999em;\"><span style=\"top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.16454285714285719em;\"><span style=\"top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.247em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.37144em;\"><span style=\"top:-2.3139999999999996em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">!<\/span><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">!<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\">&#x22EF;<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">!<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">!<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8360000000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h2 class=\"mume-header\" id=\"%E6%96%AF%E7%89%B9%E6%9E%97%E5%85%AC%E5%BC%8F\">&#x65AF;&#x7279;&#x6797;&#x516C;&#x5F0F;<\/h2>\n\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>n<\/mi><mo stretchy=\"false\">!<\/mo><mo>&#x223C;<\/mo><msqrt><mrow><mn>2<\/mn><mi>&#x3C0;<\/mi><mi>n<\/mi><\/mrow><\/msqrt><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>n<\/mi><mi>e<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mi>n<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">n!\\sim \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69444em;vertical-align:0em;\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">!<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">&#x223C;<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1.890292em;vertical-align:-0.686em;\"><\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.956095em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">&#x3C0;n<\/span><\/span><\/span><span style=\"top:-2.916095em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"\/><\/svg><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.08390500000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\"><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size2\">(<\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.10756em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">e<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size2\">)<\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.204292em;\"><span style=\"top:-3.6029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>&#x601D;&#x8DEF;<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mo>&#x222B;<\/mo><mi>k<\/mi><mrow><mi>k<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mi>ln<\/mi><mo>&#x2061;<\/mo><mi>x<\/mi><mtext>&#xA0;<\/mtext><mi mathvariant=\"bold\">d<\/mi><mi>x<\/mi><mo>&#x2212;<\/mo><mi>ln<\/mi><mo>&#x2061;<\/mo><mi>k<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>k<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mi>ln<\/mi><mo>&#x2061;<\/mo><mrow><mo fence=\"true\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mfrac><mn>1<\/mn><mi>k<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>&#x2212;<\/mo><mn>1<\/mn><mo>&#x223C;<\/mo><mfrac><mn>1<\/mn><mrow><mn>2<\/mn><mi>k<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mi>O<\/mi><mo stretchy=\"false\">(<\/mo><mfrac><mn>1<\/mn><msup><mi>k<\/mi><mn>2<\/mn><\/msup><\/mfrac><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\int_k^{k+1}\\ln{x}~\\mathbf{d}x-\\ln{k}=(k+1)\\ln\\left(1+\\frac{1}{k}\\right)-1\\sim\\frac{1}{2k}+O(\\frac{1}{k^2})<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.510958em;vertical-align:-0.9119499999999999em;\"><\/span><span class=\"mop\"><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;\">&#x222B;<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.5990080000000002em;\"><span style=\"top:-1.7880500000000001em;margin-left:-0.44445em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><span style=\"top:-3.8129000000000004em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9119499999999999em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><\/span><span class=\"mspace nobreak\">&#xA0;<\/span><span class=\"mord mathbf\">d<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.69444em;vertical-align:0em;\"><\/span><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.40003em;vertical-align:-0.95003em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">&#x223C;<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.00744em;vertical-align:-0.686em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.00744em;vertical-align:-0.686em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.740108em;\"><span style=\"top:-2.9890000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.2500em\" columnalign=\"right left\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mo>&#x2211;<\/mo><mrow><mo fence=\"true\">(<\/mo><msubsup><mo>&#x222B;<\/mo><mi>k<\/mi><mrow><mi>k<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mi>ln<\/mi><mo>&#x2061;<\/mo><mi>x<\/mi><mtext>&#xA0;<\/mtext><mi mathvariant=\"bold\">d<\/mi><mi>x<\/mi><mo>&#x2212;<\/mo><mi>ln<\/mi><mo>&#x2061;<\/mo><mi>k<\/mi><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mi>ln<\/mi><mo>&#x2061;<\/mo><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>&#x2212;<\/mo><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>&#x2212;<\/mo><mi>ln<\/mi><mo>&#x2061;<\/mo><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">!<\/mo><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>&#x223C;<\/mo><mo>&#x2211;<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>1<\/mn><mrow><mn>2<\/mn><mi>k<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mi>O<\/mi><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>1<\/mn><msup><mi>k<\/mi><mn>2<\/mn><\/msup><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>ln<\/mi><mo>&#x2061;<\/mo><mi>n<\/mi><mo>+<\/mo><mi>O<\/mi><mrow><mo fence=\"true\">(<\/mo><mn>1<\/mn><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{aligned}\\sum\\left(\\int_k^{k+1}\\ln{x}~\\mathbf{d}x-\\ln{k}\\right)&amp;=(n+1)\\ln(n+1)-(n+1)-\\ln(n!)\\\\&amp;\\sim\\sum\\left(\\frac{1}{2k}+O\\left(\\frac{1}{k^2}\\right)\\right)=\\frac{1}{2}\\ln{n}+O\\left(1\\right)\\end{aligned}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:6.0000599999999995em;vertical-align:-2.7500299999999998em;\"><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-r\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:3.2500299999999998em;\"><span style=\"top:-5.25003em;\"><span class=\"pstrut\" style=\"height:3.75em;\"><\/span><span class=\"mord\"><span class=\"mop op-symbol large-op\" style=\"position:relative;top:-0.000004999999999977245em;\">&#x2211;<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size4\">(<\/span><\/span><span class=\"mop\"><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;\">&#x222B;<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.5990080000000002em;\"><span style=\"top:-1.7880500000000001em;margin-left:-0.44445em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><span style=\"top:-3.8129000000000004em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9119499999999999em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><\/span><span class=\"mspace nobreak\">&#xA0;<\/span><span class=\"mord mathbf\">d<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size4\">)<\/span><\/span><\/span><\/span><\/span><span style=\"top:-2.2500000000000004em;\"><span class=\"pstrut\" style=\"height:3.75em;\"><\/span><span class=\"mord\"><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.7500299999999998em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:3.2500299999999998em;\"><span style=\"top:-5.25003em;\"><span class=\"pstrut\" style=\"height:3.75em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mop\">ln<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mop\">ln<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">!)<\/span><\/span><\/span><span style=\"top:-2.2500000000000004em;\"><span class=\"pstrut\" style=\"height:3.75em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">&#x223C;<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mop op-symbol large-op\" style=\"position:relative;top:-0.000004999999999977245em;\">&#x2211;<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">O<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.740108em;\"><span style=\"top:-2.9890000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">O<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose delimcenter\" style=\"top:0em;\">)<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.7500299999999998em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>&#x21D2;<\/mo><mi>ln<\/mi><mo>&#x2061;<\/mo><mrow><mi>n<\/mi><mo stretchy=\"false\">!<\/mo><\/mrow><mo>=<\/mo><mrow><mo fence=\"true\">(<\/mo><mi>n<\/mi><mo>+<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mi>ln<\/mi><mo>&#x2061;<\/mo><mi>n<\/mi><mo>&#x2212;<\/mo><mi>n<\/mi><mo>+<\/mo><mi>O<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\Rightarrow \\ln{n!}=\\left(n+\\frac{1}{2}\\right)\\ln{n}-n+O(1)<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"><\/span><span class=\"mrel\">&#x21D2;<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.69444em;vertical-align:0em;\"><\/span><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">!<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.40003em;vertical-align:-0.95003em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.66666em;vertical-align:-0.08333em;\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/p>\n<h2 class=\"mume-header\" id=\"%E9%9B%86%E7%B3%BB\">&#x96C6;&#x7CFB;<\/h2>\n\n<h2 class=\"mume-header\" id=\"%E9%80%92%E6%8E%A8\">&#x9012;&#x63A8;<\/h2>\n\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>a<\/mi><mi>n<\/mi><\/msub><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo>&#x22EF;<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">h(n)=a_nh(n-1)+\\cdots<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord mathnormal\">h<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">h<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.313em;vertical-align:0em;\"><\/span><span class=\"minner\">&#x22EF;<\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>&#x21D2;<\/mo><mfrac><mrow><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><munderover><mo>&#x220F;<\/mo><mrow><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><msub><mi>a<\/mi><mi>i<\/mi><\/msub><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><munderover><mo>&#x220F;<\/mo><mrow><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mi>a<\/mi><mi>i<\/mi><\/msub><\/mrow><\/mfrac><mo>+<\/mo><mfrac><mo lspace=\"0em\" rspace=\"0em\">&#x22EF;<\/mo><mrow><munderover><mo>&#x220F;<\/mo><mrow><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><msub><mi>a<\/mi><mi>i<\/mi><\/msub><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\Rightarrow\\frac{h(n)}{\\prod_{i=1}^na_i}=\\frac{h(n-1)}{\\prod_{i=1}^{n-1}a_i}+\\frac{\\cdots}{\\prod_{i=1}^na_i}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"><\/span><span class=\"mrel\">&#x21D2;<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.421002em;vertical-align:-0.994002em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.427em;\"><span style=\"top:-2.305708em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">&#x220F;<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.804292em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span style=\"top:-3.2029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">h<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.994002em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.5707180000000003em;vertical-align:-1.143718em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.427em;\"><span style=\"top:-2.155992em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">&#x220F;<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.954008em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span style=\"top:-3.2029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">&#x2212;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">h<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.143718em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1.984002em;vertical-align:-0.994002em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.99em;\"><span style=\"top:-2.305708em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">&#x220F;<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.804292em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span style=\"top:-3.2029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"minner\">&#x22EF;<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.994002em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h2 class=\"mume-header\" id=\"catalan-number\">Catalan Number<\/h2>\n\n<p>&#x6A21;&#x578B;&#xFF1A;&#x8FDB;&#x6808;&#x51FA;&#x6808;&#x95EE;&#x9898;&#x3001;&#x4E8C;&#x53C9;&#x6811;&#x62EC;&#x53F7;&#x5339;&#x914D;<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>C<\/mi><mn>0<\/mn><\/msub><mo>=<\/mo><mn>1<\/mn><mspace width=\"1em\"><\/mspace><msub><mi>C<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mn>1<\/mn><mspace width=\"1em\"><\/mspace><msub><mi>C<\/mi><mn>2<\/mn><\/msub><mo>=<\/mo><mn>2<\/mn><mspace width=\"1em\"><\/mspace><msub><mi>C<\/mi><mn>3<\/mn><\/msub><mo>=<\/mo><mn>5<\/mn><mspace width=\"1em\"><\/mspace><msub><mi>C<\/mi><mn>4<\/mn><\/msub><mo>=<\/mo><mn>14<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">C_0=1\\quad C_1=1\\quad C_2=2\\quad C_3=5\\quad C_4=14<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.83333em;vertical-align:-0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.83333em;vertical-align:-0.15em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.83333em;vertical-align:-0.15em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.83333em;vertical-align:-0.15em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.83333em;vertical-align:-0.15em;\"><\/span><span class=\"mord\">5<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">14<\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>C<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><munderover><mo>&#x2211;<\/mo><mrow><mi>k<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><msub><mi>C<\/mi><mrow><mi>k<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><\/mrow><\/msub><msub><mi>C<\/mi><mrow><mi>n<\/mi><mo>&#x2212;<\/mo><mi>k<\/mi><\/mrow><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">C_n=\\sum_{k=1}^nC_{k-1}C_{n-k}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.83333em;vertical-align:-0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.9535100000000005em;vertical-align:-1.302113em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.6513970000000002em;\"><span style=\"top:-1.8478869999999998em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span style=\"top:-3.0500049999999996em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">&#x2211;<\/span><\/span><\/span><span style=\"top:-4.300005em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.302113em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361079999999999em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mbin mtight\">&#x2212;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.208331em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361079999999999em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">&#x2212;<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.208331em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>&#x6839;&#x636E;&#x751F;&#x6210;&#x51FD;&#x6570;&#x6C42;&#x901A;&#x9879;<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><mo>+<\/mo><mi>x<\/mi><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">C(x)=1+xC(x)^2<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1.1141079999999999em;vertical-align:-0.25em;\"><\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>C<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><mi>n<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">C_n=\\frac{1}{n+1}\\binom{2n}{n}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.83333em;vertical-align:-0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.40003em;vertical-align:-0.95003em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7693300000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span style=\"top:-3.6769999999999996em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>&#x6839;&#x636E;&#x683C;&#x70B9;&#x56FE;&#x6C42;&#x901A;&#x9879;<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.2500em\" columnalign=\"right left\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><msub><mi>C<\/mi><mi>n<\/mi><\/msub><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mi mathvariant=\"normal\">&#x2223;<\/mi><mo stretchy=\"false\">{<\/mo><mtext>&#x5728;&#x4E3B;&#x5BF9;&#x89D2;&#x7EBF;&#x4E4B;&#x4E0A;&#x7684;&#x9053;&#x8DEF;<\/mtext><mo stretchy=\"false\">}<\/mo><mi mathvariant=\"normal\">&#x2223;<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><mi>n<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>&#x2212;<\/mo><mi mathvariant=\"normal\">&#x2223;<\/mi><mo stretchy=\"false\">{<\/mo><mtext>&#x6240;&#x6709;&#x7ECF;&#x8FC7;&#x6B21;&#x5BF9;&#x89D2;&#x7EBF;&#x7684;&#x9053;&#x8DEF;<\/mtext><mo stretchy=\"false\">}<\/mo><mi mathvariant=\"normal\">&#x2223;<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><mi>n<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>&#x2212;<\/mo><mi mathvariant=\"normal\">&#x2223;<\/mi><mo stretchy=\"false\">{<\/mo><mtext>&#x51FA;&#x53D1;&#x70B9;&#x5173;&#x4E8E;&#x6B21;&#x5BF9;&#x89D2;&#x7EBF;&#x7684;&#x5BF9;&#x79F0;&#x70B9;&#x51FA;&#x53D1;&#x7684;&#x9053;&#x8DEF;<\/mtext><mo stretchy=\"false\">}<\/mo><mi mathvariant=\"normal\">&#x2223;<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><mi>n<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>&#x2212;<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><mi>n<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{aligned}C_n&amp;=|\\{&#x5728;&#x4E3B;&#x5BF9;&#x89D2;&#x7EBF;&#x4E4B;&#x4E0A;&#x7684;&#x9053;&#x8DEF;\\}|\\\\&amp;=\\binom{2n}{n}-|\\{&#x6240;&#x6709;&#x7ECF;&#x8FC7;&#x6B21;&#x5BF9;&#x89D2;&#x7EBF;&#x7684;&#x9053;&#x8DEF;\\}|\\\\&amp;=\\binom{2n}{n}-|\\{&#x51FA;&#x53D1;&#x70B9;&#x5173;&#x4E8E;&#x6B21;&#x5BF9;&#x89D2;&#x7EBF;&#x7684;&#x5BF9;&#x79F0;&#x70B9;&#x51FA;&#x53D1;&#x7684;&#x9053;&#x8DEF;\\}|\\\\&amp;=\\binom{2n}{n}-\\binom{2n}{n+1}=\\frac{1}{n+1}\\binom{2n}{n}\\end{aligned}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:9.60009em;vertical-align:-4.550045em;\"><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-r\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:5.050045em;\"><span style=\"top:-7.660045em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-5.550044999999999em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><\/span><\/span><span style=\"top:-2.8500149999999995em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><\/span><\/span><span style=\"top:-0.14998500000000048em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:4.550045em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:5.050045em;\"><span style=\"top:-7.660045em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\">&#x2223;<\/span><span class=\"mopen\">{<\/span><span class=\"mord cjk_fallback\">&#x5728;&#x4E3B;&#x5BF9;&#x89D2;&#x7EBF;&#x4E4B;&#x4E0A;&#x7684;&#x9053;&#x8DEF;<\/span><span class=\"mclose\">}<\/span><span class=\"mord\">&#x2223;<\/span><\/span><\/span><span style=\"top:-5.550044999999999em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span style=\"top:-3.6769999999999996em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">&#x2223;<\/span><span class=\"mopen\">{<\/span><span class=\"mord cjk_fallback\">&#x6240;&#x6709;&#x7ECF;&#x8FC7;&#x6B21;&#x5BF9;&#x89D2;&#x7EBF;&#x7684;&#x9053;&#x8DEF;<\/span><span class=\"mclose\">}<\/span><span class=\"mord\">&#x2223;<\/span><\/span><\/span><span style=\"top:-2.8500149999999995em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span style=\"top:-3.6769999999999996em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">&#x2223;<\/span><span class=\"mopen\">{<\/span><span class=\"mord cjk_fallback\">&#x51FA;&#x53D1;&#x70B9;&#x5173;&#x4E8E;&#x6B21;&#x5BF9;&#x89D2;&#x7EBF;&#x7684;&#x5BF9;&#x79F0;&#x70B9;&#x51FA;&#x53D1;&#x7684;&#x9053;&#x8DEF;<\/span><span class=\"mclose\">}<\/span><span class=\"mord\">&#x2223;<\/span><\/span><\/span><span style=\"top:-0.14998500000000048em;\"><span class=\"pstrut\" style=\"height:3.45em;\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span style=\"top:-3.6769999999999996em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7693300000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7693300000000001em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span style=\"top:-3.6769999999999996em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:4.550045em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h2 class=\"mume-header\" id=\"%E9%9A%8F%E6%9C%BA%E6%B8%B8%E8%B5%B0\">&#x968F;&#x673A;&#x6E38;&#x8D70;<\/h2>\n\n<h2 class=\"mume-header\" id=\"%E5%AE%B9%E6%96%A5%E5%8E%9F%E7%90%86\">&#x5BB9;&#x65A5;&#x539F;&#x7406;<\/h2>\n\n<h2 class=\"mume-header\" id=\"%E9%94%99%E6%8E%92\">&#x9519;&#x6392;<\/h2>\n\n<p>&#x9519;&#x6392;&#x6982;&#x7387;&#x8D8B;&#x4E8E;<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mn>1<\/mn><mi>e<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1}{e}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.190108em;vertical-align:-0.345em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.845108em;\"><span style=\"top:-2.6550000000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">e<\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.345em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>&#x9001;&#x5BF9;&#x4FE1;&#x5C01;&#x671F;&#x671B;<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>E<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">E(X)=1<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/p>\n<h2 class=\"mume-header\" id=\"stirling-number\">Stirling Number<\/h2>\n\n<h3 class=\"mume-header\" id=\"1st-kind\">1st kind<\/h3>\n\n<p>n&#x4E2A;&#x6570;&#x7684;&#x7F6E;&#x6362;&#x4E2D;&#x6070;&#x6709;m&#x4E2A;&#x5708;&#x7684;&#x4E2A;&#x6570;<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><mi>o<\/mi><mi>r<\/mi><mspace width=\"1em\"><\/mspace><mrow><mo fence=\"true\">[<\/mo><mfrac linethickness=\"0px\"><mi>n<\/mi><mi>m<\/mi><\/mfrac><mo fence=\"true\">]<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">S_1(n,m)\\quad or\\quad {n\\brack m}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.40003em;vertical-align:-0.95003em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">or<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">[<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.1075599999999999em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">m<\/span><\/span><\/span><span style=\"top:-3.6769999999999996em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>S<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><msub><mi>S<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">S_1(n,m)=S_1(n-1,m-1)+(n-1)S_1(n-1,m)<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.8388800000000001em;vertical-align:-0.19444em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>x<\/mi><mover accent=\"true\"><mi>n<\/mi><mo stretchy=\"true\">&#x203E;<\/mo><\/mover><\/msup><mo>=<\/mo><munderover><mo>&#x2211;<\/mo><mrow><mi>m<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><mi>n<\/mi><\/munderover><msub><mi>S<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo stretchy=\"false\">)<\/mo><msup><mi>x<\/mi><mi>m<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">x^{\\overline{n}}=\\sum_{m=0}^nS_1(n,m)x^m<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8858920000000001em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8858920000000001em;\"><span style=\"top:-3.413em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord overline mtight\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6755600000000002em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><span style=\"top:-3.57756em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"overline-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.9185100000000004em;vertical-align:-1.267113em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.6513970000000002em;\"><span style=\"top:-1.882887em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span style=\"top:-3.050005em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">&#x2211;<\/span><\/span><\/span><span style=\"top:-4.3000050000000005em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.267113em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mclose\">)<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7143919999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>x<\/mi><munder accentunder=\"true\"><mi>n<\/mi><mo stretchy=\"true\">&#x203E;<\/mo><\/munder><\/msup><mo>=<\/mo><munderover><mo>&#x2211;<\/mo><mrow><mi>m<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><mi>n<\/mi><\/munderover><mo stretchy=\"false\">(<\/mo><mo>&#x2212;<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mrow><mi>n<\/mi><mo>+<\/mo><mi>m<\/mi><\/mrow><\/msup><msub><mi>S<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo stretchy=\"false\">)<\/mo><msup><mi>x<\/mi><mi>m<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">x^{\\underline{n}}=\\sum_{m=0}^n(-1)^{n+m}S_1(n,m)x^m<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7143919999999999em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7143919999999999em;\"><span style=\"top:-3.413em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord underline mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.43056em;\"><span style=\"top:-2.804em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"underline-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.245em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.9185100000000004em;vertical-align:-1.267113em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.6513970000000002em;\"><span style=\"top:-1.882887em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span style=\"top:-3.050005em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">&#x2211;<\/span><\/span><\/span><span style=\"top:-4.3000050000000005em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.267113em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">&#x2212;<\/span><span class=\"mord\">1<\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.821331em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mclose\">)<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7143919999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h3 class=\"mume-header\" id=\"2nd-kind\">2nd kind<\/h3>\n\n<p>n&#x4E2A;&#x4E0D;&#x540C;&#x7403;&#x653E;&#x5230;m&#x4E2A;&#x76F8;&#x540C;&#x76D2;&#xFF0C;&#x6EE1;&#x8DB3;&#x6240;&#x6709;&#x76D2;&#x5B50;&#x4E0D;&#x7A7A;&#x7684;&#x65B9;&#x6CD5;&#x6570;<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><mi>o<\/mi><mi>r<\/mi><mspace width=\"1em\"><\/mspace><mrow><mo fence=\"true\">{<\/mo><mfrac linethickness=\"0px\"><mi>n<\/mi><mi>m<\/mi><\/mfrac><mo fence=\"true\">}<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">S_2(n,m)\\quad or\\quad {n\\brace m}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.40003em;vertical-align:-0.95003em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">or<\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">{<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.1075599999999999em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">m<\/span><\/span><\/span><span style=\"top:-3.6769999999999996em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">}<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>S<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>m<\/mi><msub><mi>S<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">S_2(n,m)=S_2(n-1,m-1)+mS_2(n-1,m)<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.8388800000000001em;vertical-align:-0.19444em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>x<\/mi><mi>n<\/mi><\/msup><mo>=<\/mo><munderover><mo>&#x2211;<\/mo><mrow><mi>m<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><mi>n<\/mi><\/munderover><msub><mi>S<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo stretchy=\"false\">)<\/mo><msup><mi>x<\/mi><munder accentunder=\"true\"><mi>m<\/mi><mo stretchy=\"true\">&#x203E;<\/mo><\/munder><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">x^n=\\sum_{m=0}^nS_2(n,m)x^{\\underline{m}}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7143919999999999em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7143919999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.9185100000000004em;vertical-align:-1.267113em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.6513970000000002em;\"><span style=\"top:-1.882887em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span style=\"top:-3.050005em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">&#x2211;<\/span><\/span><\/span><span style=\"top:-4.3000050000000005em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.267113em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mclose\">)<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7143919999999999em;\"><span style=\"top:-3.413em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord underline mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.43056em;\"><span style=\"top:-2.804em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"underline-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.245em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>x<\/mi><mi>n<\/mi><\/msup><mo>=<\/mo><munderover><mo>&#x2211;<\/mo><mrow><mi>m<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><mi>n<\/mi><\/munderover><mo stretchy=\"false\">(<\/mo><mo>&#x2212;<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mrow><mi>n<\/mi><mo>+<\/mo><mi>m<\/mi><\/mrow><\/msup><msub><mi>S<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo stretchy=\"false\">)<\/mo><msup><mi>x<\/mi><mover accent=\"true\"><mi>m<\/mi><mo stretchy=\"true\">&#x203E;<\/mo><\/mover><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">x^n=\\sum_{m=0}^n(-1)^{n+m}S_2(n,m)x^{\\overline{m}}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7143919999999999em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7143919999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:2.9185100000000004em;vertical-align:-1.267113em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.6513970000000002em;\"><span style=\"top:-1.882887em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span style=\"top:-3.050005em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span><span class=\"mop op-symbol large-op\">&#x2211;<\/span><\/span><\/span><span style=\"top:-4.3000050000000005em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.267113em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">&#x2212;<\/span><span class=\"mord\">1<\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.821331em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mclose\">)<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8858920000000001em;\"><span style=\"top:-3.413em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord overline mtight\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6755600000000002em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><span style=\"top:-3.57756em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"overline-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h2 class=\"mume-header\" id=\"%E7%90%83%E7%9B%92%E9%97%AE%E9%A2%98\">&#x7403;&#x76D2;&#x95EE;&#x9898;<\/h2>\n\n<p>n&#x4E2A;&#x7403;&#xFF0C;m&#x4E2A;&#x76D2;<\/p>\n<table>\n<thead>\n<tr>\n<th style=\"text-align:center\">&#x7403;&#x540C;&#x5426;<\/th>\n<th style=\"text-align:center\">&#x76D2;&#x540C;&#x5426;<\/th>\n<th style=\"text-align:center\">&#x7A7A;&#x5426;<\/th>\n<th style=\"text-align:left\">&#x6570;&#x91CF;<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align:center\">&#x7403;&#x4E0D;&#x540C;<\/td>\n<td style=\"text-align:center\">&#x76D2;&#x4E0D;&#x540C;<\/td>\n<td style=\"text-align:center\">&#x4E0D;&#x53EF;&#x7A7A;<\/td>\n<td style=\"text-align:left\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>m<\/mi><mi>n<\/mi><\/msup><mo>&#x2212;<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>m<\/mi><mn>1<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mi>m<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mi>n<\/mi><\/msup><mo>+<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>m<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mi>m<\/mi><mo>&#x2212;<\/mo><mn>2<\/mn><msup><mo stretchy=\"false\">)<\/mo><mi>n<\/mi><\/msup><mo>&#x2212;<\/mo><mo>&#x22EF;<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">m^n-\\binom{m}{1}(m-1)^n+\\binom{m}{2}(m-2)^n-\\cdots<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.747722em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">m<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.664392em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1.20001em;vertical-align:-0.35001em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7453919999999999em;\"><span style=\"top:-2.3550000000000004em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span style=\"top:-3.144em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.345em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.664392em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1.20001em;vertical-align:-0.35001em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7453919999999999em;\"><span style=\"top:-2.3550000000000004em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span style=\"top:-3.144em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.345em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.664392em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.313em;vertical-align:0em;\"><\/span><span class=\"minner\">&#x22EF;<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:center\">&#x7403;&#x4E0D;&#x540C;<\/td>\n<td style=\"text-align:center\">&#x76D2;&#x4E0D;&#x540C;<\/td>\n<td style=\"text-align:center\">&#x53EF;&#x7A7A;<\/td>\n<td style=\"text-align:left\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>m<\/mi><mi>n<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">m^n<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.664392em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">m<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.664392em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:center\">&#x7403;&#x4E0D;&#x540C;<\/td>\n<td style=\"text-align:center\">&#x76D2;&#x540C;<\/td>\n<td style=\"text-align:center\">&#x4E0D;&#x53EF;&#x7A7A;<\/td>\n<td style=\"text-align:left\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>S<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mi mathvariant=\"normal\">.<\/mi><mi>m<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><msup><mi>m<\/mi><mi>n<\/mi><\/msup><mo>&#x2212;<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>m<\/mi><mn>1<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mi>m<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mi>n<\/mi><\/msup><mo>+<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>m<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mi>m<\/mi><mo>&#x2212;<\/mo><mn>2<\/mn><msup><mo stretchy=\"false\">)<\/mo><mi>n<\/mi><\/msup><mo>&#x2212;<\/mo><mo>&#x22EF;<\/mo><\/mrow><mrow><mi>m<\/mi><mo stretchy=\"false\">!<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">S_2(n.m)=\\frac{m^n-\\binom{m}{1}(m-1)^n+\\binom{m}{2}(m-2)^n-\\cdots}{m!}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">.<\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1.8550099999999998em;vertical-align:-0.345em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.5100099999999999em;\"><span style=\"top:-2.6550000000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mclose mtight\">!<\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.71251em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7385428571428572em;\"><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mbin mtight\">&#x2212;<\/span><span class=\"mord mtight\"><span class=\"mopen sizing reset-size3 size6 mtight delimcenter\" style=\"top:0.07500000000000001em;\"><span class=\"delimsizing size1 mtight\"><span class=\"mtight\">(<\/span><\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7785428571428572em;\"><span style=\"top:-2.156em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span style=\"top:-2.971em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose sizing reset-size3 size6 mtight delimcenter\" style=\"top:0.07500000000000001em;\"><span class=\"delimsizing size1 mtight\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mbin mtight\">&#x2212;<\/span><span class=\"mord mtight\">1<\/span><span class=\"mclose mtight\"><span class=\"mclose mtight\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7385428571428572em;\"><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mtight\"><span class=\"mopen sizing reset-size3 size6 mtight delimcenter\" style=\"top:0.07500000000000001em;\"><span class=\"delimsizing size1 mtight\"><span class=\"mtight\">(<\/span><\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7785428571428572em;\"><span style=\"top:-2.156em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span style=\"top:-2.971em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose sizing reset-size3 size6 mtight delimcenter\" style=\"top:0.07500000000000001em;\"><span class=\"delimsizing size1 mtight\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mbin mtight\">&#x2212;<\/span><span class=\"mord mtight\">2<\/span><span class=\"mclose mtight\"><span class=\"mclose mtight\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7385428571428572em;\"><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mbin mtight\">&#x2212;<\/span><span class=\"minner mtight\">&#x22EF;<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.345em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:center\">&#x7403;&#x4E0D;&#x540C;<\/td>\n<td style=\"text-align:center\">&#x76D2;&#x540C;<\/td>\n<td style=\"text-align:center\">&#x53EF;&#x7A7A;<\/td>\n<td style=\"text-align:left\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>S<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>S<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>S<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo>&#x2212;<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo>&#x22EF;<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">S_2(n,m)+S_2(n,m-1)+S_2(n,m-2)+\\cdots<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">&#x2212;<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.313em;vertical-align:0em;\"><\/span><span class=\"minner\">&#x22EF;<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:center\">&#x7403;&#x540C;<\/td>\n<td style=\"text-align:center\">&#x76D2;&#x4E0D;&#x540C;<\/td>\n<td style=\"text-align:center\">&#x4E0D;&#x53EF;&#x7A7A;<\/td>\n<td style=\"text-align:left\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mi>n<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><\/mrow><mrow><mi>m<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\binom{n-1}{m-1}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.2984390000000001em;vertical-align:-0.403331em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.895108em;\"><span style=\"top:-2.355em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mbin mtight\">&#x2212;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span style=\"top:-3.144em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">&#x2212;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.403331em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:center\">&#x7403;&#x540C;<\/td>\n<td style=\"text-align:center\">&#x76D2;&#x4E0D;&#x540C;<\/td>\n<td style=\"text-align:center\">&#x53EF;&#x7A7A;<\/td>\n<td style=\"text-align:left\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mrow><mi>n<\/mi><mo>+<\/mo><mi>m<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><\/mrow><mrow><mi>m<\/mi><mo>&#x2212;<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\binom{n+m-1}{m-1}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.2984390000000001em;vertical-align:-0.403331em;\"><\/span><span class=\"mord\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.895108em;\"><span style=\"top:-2.355em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mbin mtight\">&#x2212;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span style=\"top:-3.144em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mbin mtight\">&#x2212;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">&#x200B;<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.403331em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:center\">&#x7403;&#x540C;<\/td>\n<td style=\"text-align:center\">&#x76D2;&#x540C;<\/td>\n<td style=\"text-align:center\">&#x4E0D;&#x53EF;&#x7A7A;<\/td>\n<td style=\"text-align:left\"><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align:center\">&#x7403;&#x540C;<\/td>\n<td style=\"text-align:center\">&#x76D2;&#x540C;<\/td>\n<td style=\"text-align:center\">&#x53EF;&#x7A7A;<\/td>\n<td style=\"text-align:left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n      <\/div>\n      \n      \n    \n    \n    \n    \n    \n    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