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CatalanNumber






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > CatalanNumber[n] > Product representations





http://functions.wolfram.com/06.41.08.0002.01









  


  










Input Form





CatalanNumber[z] == ((2^(2 z + 1) E^(EulerGamma (2 + z)) (2 + z))/ (Sqrt[Pi] (1 + 2 z))) Product[(((1 + 1/k)^(z + 1/2)/(1 + (2 z + 1)/(2 k))) (1 + (z + 2)/k))/ E^((z + 2)/k), {k, 1, Infinity}] /; !(Element[-z - 1/2, Integers] && -z - 1/2 >= 0)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["CatalanNumber", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", "z"]], "+", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["EulerGamma", " ", RowBox[List["(", RowBox[List["2", "+", "z"]], ")"]]]]], " ", RowBox[List["(", RowBox[List["2", "+", "z"]], ")"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", "z"]]]], ")"]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox["1", "k"]]], ")"]], RowBox[List["z", "+", FractionBox["1", "2"]]]], RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["2", "z"]], "+", "1"]], RowBox[List["2", "k"]]]]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["z", "+", "2"]], "k"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["z", "+", "2"]], "k"]]]]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[RowBox[List["Element", "[", RowBox[List[RowBox[List[RowBox[List["-", "z"]], "-", FractionBox["1", "2"]]], ",", "Integers"]], "]"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", "z"]], "-", FractionBox["1", "2"]]], "\[GreaterEqual]", "0"]]]], "]"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, CatalanNumber] </annotation> </semantics> <mi> z </mi> </msub> <mo> &#63449; </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> k </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mi> k </mi> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mi> k </mi> </mfrac> </mrow> </msup> </mrow> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> CatalanNumber </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <eulergamma /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["CatalanNumber", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "z"]], "+", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["EulerGamma", " ", RowBox[List["(", RowBox[List["2", "+", "z"]], ")"]]]]], " ", RowBox[List["(", RowBox[List["2", "+", "z"]], ")"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox["1", "k"]]], ")"]], RowBox[List["z", "+", FractionBox["1", "2"]]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["z", "+", "2"]], "k"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["z", "+", "2"]], "k"]]]]]], RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["2", " ", "z"]], "+", "1"]], RowBox[List["2", " ", "k"]]]]]]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "z"]]]], ")"]]]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "z"]], "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", "z"]], "-", FractionBox["1", "2"]]], "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]










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





2007-05-02





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