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http://functions.wolfram.com/06.01.27.0003.01
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n! == 2^((1/4) (Cos[2 n Pi] - 1) - n) Pi^((1/2) Sin[n Pi]^2) (2 n)!!
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Cell[BoxData[RowBox[List[RowBox[List["n", "!"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", "n", " ", "\[Pi]"]], "]"]], "-", "1"]], ")"]]]], "-", "n"]]], " ", SuperscriptBox["\[Pi]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["n", " ", "\[Pi]"]], "]"]], "2"]]]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "n"]], ")"]], "!!"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> <pi /> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <pi /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> n </ci> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> Factorial2 </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["n_", "!"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", "n", " ", "\[Pi]"]], "]"]], "-", "1"]], ")"]]]], "-", "n"]]], " ", SuperscriptBox["\[Pi]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["n", " ", "\[Pi]"]], "]"]], "2"]]]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "n"]], ")"]], "!!"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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