CatalanNumber[n + 1/2] == (2^(3 n + 3) n!)/ (Pi Product[2 k - 1, {k, 1, n + 2}]) /; Element[n, Integers] && n >= 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["CatalanNumber", "[", RowBox[List["n", "+", FractionBox["1", "2"]]], "]"]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["3", " ", "n"]], "+", "3"]]], " ", RowBox[List["n", "!"]]]], RowBox[List["\[Pi]", " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], RowBox[List["n", "+", "2"]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]], ")"]]]]]]]]], "/;", " ", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

<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> <mrow> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> &#63449; </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> CatalanNumber </ci> <apply> <plus /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </uplimit> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["CatalanNumber", "[", RowBox[List["n_", "+", FractionBox["1", "2"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["3", " ", "n"]], "+", "3"]]], " ", RowBox[List["n", "!"]]]], RowBox[List["\[Pi]", " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], RowBox[List["n", "+", "2"]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]], ")"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

2007-05-02