CatalanNumber[x] \[Proportional] 4^x/(E^(9/(8 x)) (Sqrt[Pi] x^(3/2))) /; (x -> Infinity)

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<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> x </mi> </msub> <mo> &#8733; </mo> <mfrac> <mrow> <msup> <mn> 4 </mn> <mi> x </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mn> 9 </mn> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> </mfrac> </mrow> </msup> </mrow> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msup> <mi> x </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> CatalanNumber </ci> <ci> x </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <ci> x </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> x </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> x </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> x </ci> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["CatalanNumber", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["4", "x"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["9", RowBox[List["8", " ", "x"]]]]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox["x", RowBox[List["3", "/", "2"]]]]]], "/;", RowBox[List["(", RowBox[List["x", "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

2007-05-02