Double factorial: Series representations (formula 06.02.06.0009)

Factorial2

$Mathematica Notation$

Gamma, Beta, Erf

Factorial2[n]

Series representations

Asymptotic series expansions

http://functions.wolfram.com/06.02.06.0009.01

Input Form

n!! \[Proportional] ((2/Pi)^((1/4) (1 - Cos[Pi n])) Sqrt[Pi] n^((n + 1)/2) (1 + Sum[((-1)^j PermutationCyclesD[2 (k + j), j])/(2^j (k + j)!)/n^k, {k, 1, Infinity}, {j, 1, 2 k}]))/E^(n/2) /; (Abs[Arg[n]] < Pi && (Abs[n] -> Infinity) && PermutationCyclesD[m, j] == (m - 1) (PermutationCyclesD[m - 1, j] + (m - 2) PermutationCyclesD[m - 3, j - 1]) && PermutationCyclesD[0, 0] == 1 && PermutationCyclesD[m, 1] == (m - 1)! && PermutationCyclesD[m, j] == 0 /; m <= 3 j - 1)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["n", "!!"]], "\[Proportional]", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "\[Pi]"], ")"]], RowBox[List[FractionBox["1", "4"], RowBox[List["(", RowBox[List["1", "-", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "n"]], "]"]]]], ")"]]]]], SqrtBox["\[Pi]"], " ", SuperscriptBox["n", FractionBox[RowBox[List["n", "+", "1"]], "2"]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "n"]], "/", "2"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["2", " ", "k"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["PermutationCyclesD", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", "j"]], ")"]]]], ",", "j"]], "]"]]]], RowBox[List[SuperscriptBox["2", "j"], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "j"]], ")"]], "!"]]]]], SuperscriptBox["n", RowBox[List["-", "k"]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "n", "]"]], "]"]], "<", "\[Pi]"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "n", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List[RowBox[List["PermutationCyclesD", "[", RowBox[List["m", ",", "j"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["PermutationCyclesD", "[", RowBox[List[RowBox[List["m", "-", "1"]], ",", "j"]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "2"]], ")"]], RowBox[List["PermutationCyclesD", "[", RowBox[List[RowBox[List["m", "-", "3"]], ",", RowBox[List["j", "-", "1"]]]], "]"]]]]]], ")"]]]]]], "\[And]", RowBox[List[RowBox[List["PermutationCyclesD", "[", RowBox[List["0", ",", "0"]], "]"]], "\[Equal]", "1"]], "\[And]", "\[IndentingNewLine]", RowBox[List[RowBox[List["PermutationCyclesD", "[", RowBox[List["m", ",", "1"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]]], "\[And]", "\[IndentingNewLine]", RowBox[List[RowBox[List["PermutationCyclesD", "[", RowBox[List["m", ",", "j"]], "]"]], "\[Equal]", "0"]]]], "/;", " ", RowBox[List["m", "\[LessEqual]", RowBox[List[RowBox[List["3", "j"]], "-", "1"]]]]]], ")"]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> n </mi> <mo> !! </mo> </mrow> <mo> ∝ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <msup> <mi> n </mi> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> P </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> n </mi> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> <mrow> <msup> <mn> 2 </mn> <mi> j </mi> </msup> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mi> π </mi> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> n </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> P </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> P </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> , </mo> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> P </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> P </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> P </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> P </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> ≤ </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Factorial2 </ci> <ci> n </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <apply> <times /> <pi /> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> n </ci> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </uplimit> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <ci> P </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> <ci> j </ci> </apply> <apply> <power /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <and /> <apply> <lt /> <apply> <abs /> <apply> <arg /> <ci> n </ci> </apply> </apply> <pi /> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> n </ci> </apply> <infinity /> </apply> <apply> <eq /> <apply> <ci> P </ci> <ci> m </ci> <ci> j </ci> </apply> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> <apply> <ci> P </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -3 </cn> </apply> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> P </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <ci> j </ci> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> P </ci> <cn type='integer'> 0 </cn> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <ci> P </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> P </ci> <ci> m </ci> <ci> j </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <leq /> <ci> m </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29

Factorial[n]