n!! == (n + 2 m)!!/Pochhammer[1 + n/2, m]/2^m /; Element[m, Integers]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["n", "!!"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "+", RowBox[List["2", "m"]]]], ")"]], "!!"]], RowBox[List[" ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", FractionBox["n", "2"]]], ",", "m"]], "]"]]]]]]]]], "/;", RowBox[List["m", "\[Element]", "Integers"]]]]]]

<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> &#10869; </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> </mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[FractionBox[&quot;n&quot;, &quot;2&quot;], &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;m&quot;], Pochhammer] </annotation> </semantics> </mfrac> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Factorial2 </ci> <ci> n </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <ci> Factorial2 </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["n_", "!!"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", RowBox[List["2", " ", "m"]]]], ")"]], "!!"]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", FractionBox["n", "2"]]], ",", "m"]], "]"]]], "/;", RowBox[List["m", "\[Element]", "Integers"]]]]]]]]

2001-10-29