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variants of this functions
Factorial2






Mathematica Notation

Traditional 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





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MathML Form







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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





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