Double factorial: Series representations (formula 06.02.06.0001)

Factorial2

$Mathematica Notation$

Gamma, Beta, Erf

Factorial2[n]

Series representations

Generalized power series

Expansions at n==n₀/;n₀!=-2m

http://functions.wolfram.com/06.02.06.0001.02

Input Form

n!! \[Proportional] Subscript[n, 0]!! (1 + (1/4) (Log[4] + 2 PolyGamma[1 + Subscript[n, 0]/2] + Pi Log[2/Pi] Sin[Subscript[n, 0] Pi]) (n - Subscript[n, 0]) + (1/32) (4 Log[2]^2 + 4 PolyGamma[1 + Subscript[n, 0]/2]^2 + 4 PolyGamma[1, 1 + Subscript[n, 0]/2] + 4 PolyGamma[1 + Subscript[n, 0]/2] (2 Log[2] + Pi Log[2/Pi] Sin[Subscript[n, 0] Pi]) + Pi Log[2/Pi] (4 Pi Cos[Subscript[n, 0] Pi] + Sin[Subscript[n, 0] Pi] (4 Log[2] + Pi Log[2/Pi] Sin[Subscript[n, 0] Pi]))) (n - Subscript[n, 0])^2 + (1/384) (8 Log[2]^3 + 8 PolyGamma[1 + Subscript[n, 0]/2]^3 + 8 PolyGamma[2, 1 + Subscript[n, 0]/2] + 12 (PolyGamma[1 + Subscript[n, 0]/2]^2 + PolyGamma[1, 1 + Subscript[n, 0]/2]) (Log[4] + Pi Log[2/Pi] Sin[Pi Subscript[n, 0]]) + 6 PolyGamma[1 + Subscript[n, 0]/2] (4 Log[2]^2 + 4 PolyGamma[1, 1 + Subscript[n, 0]/2] + Pi Log[2/Pi] (4 Pi Cos[Subscript[n, 0] Pi] + Sin[Subscript[n, 0] Pi] (Log[16] + Pi Log[2/Pi] Sin[Subscript[n, 0] Pi]))) + Pi Log[2/Pi] (24 Pi Cos[Subscript[n, 0] Pi] Log[2] + Sin[Subscript[n, 0] Pi] (-16 Pi^2 + 12 Log[2]^2 + Pi Log[2/Pi] Sin[Subscript[n, 0] Pi] (Log[64] + Pi Log[2/Pi] Sin[Subscript[n, 0] Pi])) + 6 Pi^2 Log[2/Pi] Sin[2 Subscript[n, 0] Pi])) (n - Subscript[n, 0])^3 + \[Ellipsis]) /; (n -> Subscript[n, 0]) && !(Element[-(Subscript[n, 0]/2), Integers] && -(Subscript[n, 0]/2) > 0)

Standard Form

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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> <mrow> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> !! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> <mo> ) 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<mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <msub> <mi> n </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 384 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <msup> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 16 </mn> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 24 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 16 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 64 </mn> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> n </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <msub> <mi> n </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mo> … </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <msub> <mi> n </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mn> 2 </mn> </mfrac> </mrow> <mo> ∉ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </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> <ci> Factorial2 </ci> <apply> <ci> Subscript 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</cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> <apply> <cos /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <pi /> </apply> </apply> </apply> <apply> <times /> <apply> <sin /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <pi /> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <pi /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <pi /> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> 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type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> 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/> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 24 </cn> <pi /> <apply> <cos /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <pi /> </apply> </apply> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <sin /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <pi /> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <pi /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <pi /> </apply> </apply> <apply> <plus /> <apply> <ln /> <cn type='integer'> 64 </cn> </apply> <apply> <times /> <pi /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <pi /> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript 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</annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29

Factorial[n]