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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/06.41.16.0007.01









  


  








Input Form





CatalanNumber[z] CatalanNumber[1 - z] == (2 (1 - 2 z) Tan[Pi z])/ (Pi (z - 2) (z - 1) z (z + 1))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["CatalanNumber", "[", "z", "]"]], " ", RowBox[List["CatalanNumber", "[", RowBox[List["1", "-", "z"]], "]"]]]], "\[Equal]", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "z"]]]], ")"]], " ", RowBox[List["Tan", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["z", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], " ", "z", " ", RowBox[List["(", RowBox[List["z", "+", "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> <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> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msub> </mrow> <mo> &#63449; </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <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> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <ci> CatalanNumber </ci> <ci> z </ci> </apply> <apply> <ci> CatalanNumber </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </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'> -2 </cn> </apply> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </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["1", "-", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "z"]]]], ")"]], " ", RowBox[List["Tan", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["z", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], " ", "z", " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]]]]]]]


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





2007-05-02