Double factorial: Series representations (formula 06.02.06.0008)

Factorial2

$Mathematica Notation$

Gamma, Beta, Erf

Factorial2[n]

Series representations

Asymptotic series expansions

http://functions.wolfram.com/06.02.06.0008.01

Input Form

n!! \[Proportional] ((2/Pi)^((1/4) (1 - Cos[Pi n])) Sqrt[Pi] n^((n + 1)/2) (1 + 1/(6 n) + 1/(72 n^2) - 139/(6480 n^3) - 571/(155520 n^4) + 163879/(6531840 n^5) + 5246819/(1175731200 n^6) - 534703531/(7054387200 n^7) - 4483131259/(338610585600 n^8) + 432261921612371/(1005673439232000 n^9) + O[1/n^10]))/E^(n/2) /; Abs[Arg[n]] < Pi && (Abs[n] -> Infinity)

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", "+", FractionBox["1", RowBox[List["6", " ", "n"]]], "+", FractionBox["1", RowBox[List["72", " ", SuperscriptBox["n", "2"]]]], "-", FractionBox["139", RowBox[List["6480", " ", SuperscriptBox["n", "3"]]]], "-", FractionBox["571", RowBox[List["155520", " ", SuperscriptBox["n", "4"]]]], "+", FractionBox["163879", RowBox[List["6531840", " ", SuperscriptBox["n", "5"]]]], "+", FractionBox["5246819", RowBox[List["1175731200", " ", SuperscriptBox["n", "6"]]]], "-", FractionBox["534703531", RowBox[List["7054387200", " ", SuperscriptBox["n", "7"]]]], "-", FractionBox["4483131259", RowBox[List["338610585600", " ", SuperscriptBox["n", "8"]]]], "+", FractionBox["432261921612371", RowBox[List["1005673439232000", " ", SuperscriptBox["n", "9"]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["n", "10"]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "n", "]"]], "]"]], "<", "\[Pi]"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "n", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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> <mfrac> <mn> 1 </mn> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 72 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 139 </mn> <mrow> <mn> 6480 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 571 </mn> <mrow> <mn> 155520 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 163879 </mn> <mrow> <mn> 6531840 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 5 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 5246819 </mn> <mrow> <mn> 1175731200 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 6 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 534703531 </mn> <mrow> <mn> 7054387200 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 7 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 4483131259 </mn> <mrow> <mn> 338610585600 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 8 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 432261921612371 </mn> <mrow> <mn> 1005673439232000 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 9 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msup> <mi> n </mi> <mn> 10 </mn> </msup> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <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> </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> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 72 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 139 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 6480 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 571 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 155520 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 163879 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 6531840 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 5 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5246819 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 1175731200 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 6 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 534703531 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 7054387200 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 7 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4483131259 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 338610585600 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 8 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 432261921612371 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 1005673439232000 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 9 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> n </ci> <cn type='integer'> 10 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <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> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["n_", "!!"]], "]"]], "\[RuleDelayed]", RowBox[List[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["-", FractionBox["n", "2"]]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox["1", RowBox[List["6", " ", "n"]]], "+", FractionBox["1", RowBox[List["72", " ", SuperscriptBox["n", "2"]]]], "-", FractionBox["139", RowBox[List["6480", " ", SuperscriptBox["n", "3"]]]], "-", FractionBox["571", RowBox[List["155520", " ", SuperscriptBox["n", "4"]]]], "+", FractionBox["163879", RowBox[List["6531840", " ", SuperscriptBox["n", "5"]]]], "+", FractionBox["5246819", RowBox[List["1175731200", " ", SuperscriptBox["n", "6"]]]], "-", FractionBox["534703531", RowBox[List["7054387200", " ", SuperscriptBox["n", "7"]]]], "-", FractionBox["4483131259", RowBox[List["338610585600", " ", SuperscriptBox["n", "8"]]]], "+", FractionBox["432261921612371", RowBox[List["1005673439232000", " ", SuperscriptBox["n", "9"]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["n", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "10"]], "]"]]]], ")"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "n", "]"]], "]"]], "<", "\[Pi]"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "n", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]]]

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

2001-10-29

Factorial[n]