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CatalanNumber




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > CatalanNumber[n] > Transformations > Products, sums, and powers of the direct function > Products of the direct function





http://functions.wolfram.com/06.41.16.0008.01









  


  








Input Form





CatalanNumber[z] CatalanNumber[-z - n] == -((Pochhammer[1 + z, n - 2] Tan[Pi z])/ (4^n (Pi (1 + z) Pochhammer[1/2 + z, n]))) /; Element[n, Integers]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["CatalanNumber", "[", "z", "]"]], RowBox[List["CatalanNumber", "[", RowBox[List[RowBox[List["-", "z"]], "-", "n"]], "]"]]]], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["4", RowBox[List["-", "n"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "z"]], ",", RowBox[List["n", "-", "2"]]]], "]"]], " ", RowBox[List["Tan", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "z"]], ",", "n"]], "]"]]]]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]








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> <msub> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, CatalanNumber] </annotation> </semantics> <mi> z </mi> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, CatalanNumber] </annotation> </semantics> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> z </mi> </mrow> </msub> </mrow> <mo> &#63449; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mn> 4 </mn> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;z&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[&quot;n&quot;, &quot;-&quot;, &quot;2&quot;]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> tan </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;z&quot;, &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], &quot;)&quot;]], &quot;n&quot;], Pochhammer] </annotation> </semantics> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <ci> CatalanNumber </ci> <ci> z </ci> </apply> <apply> <ci> CatalanNumber </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <tan /> <apply> <times /> <pi /> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["CatalanNumber", "[", "z_", "]"]], " ", RowBox[List["CatalanNumber", "[", RowBox[List[RowBox[List["-", "z_"]], "-", "n_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["4", RowBox[List["-", "n"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "z"]], ",", RowBox[List["n", "-", "2"]]]], "]"]], " ", RowBox[List["Tan", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "z"]], ",", "n"]], "]"]]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]]]


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





2007-05-02