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variants of this functions
Factorial2




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Factorial2[n] > Identities > Functional identities > Relations of special kind





http://functions.wolfram.com/06.02.17.0005.01









  


  








Input Form





f[n] == n f[n - 2] /; f[n] == n!! g[n] && g[n] == g[n - 2] && f[1] == 1








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["f", "[", "n", "]"]], "\[Equal]", RowBox[List["n", " ", RowBox[List["f", "[", RowBox[List["n", "-", "2"]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["f", "[", "n", "]"]], "\[Equal]", RowBox[List[RowBox[List["n", "!!"]], RowBox[List["g", "[", "n", "]"]]]]]], "\[And]", RowBox[List[RowBox[List["g", "[", "n", "]"]], "\[Equal]", RowBox[List["g", "[", RowBox[List["n", "-", "2"]], "]"]]]], "\[And]", RowBox[List[RowBox[List["f", "[", "1", "]"]], "\[Equal]", "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> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> n </mi> <mo> !! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> f </ci> <ci> n </ci> </apply> <apply> <times /> <ci> n </ci> <apply> <ci> f </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> f </ci> <ci> n </ci> </apply> <apply> <times /> <apply> <ci> Factorial2 </ci> <ci> n </ci> </apply> <apply> <ci> g </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> g </ci> <ci> n </ci> </apply> <apply> <ci> g </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> f </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["f", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["n", " ", RowBox[List["f", "[", RowBox[List["n", "-", "2"]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["f", "[", "n", "]"]], "\[Equal]", RowBox[List[RowBox[List["n", "!!"]], " ", RowBox[List["g", "[", "n", "]"]]]]]], "&&", RowBox[List[RowBox[List["g", "[", "n", "]"]], "\[Equal]", RowBox[List["g", "[", RowBox[List["n", "-", "2"]], "]"]]]], "&&", RowBox[List[RowBox[List["f", "[", "1", "]"]], "\[Equal]", "1"]]]]]]]]]]


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





2001-10-29