BellB[n] \[Proportional] ((n!/Sqrt[2 Pi ProductLog[n]^2 Exp[ProductLog[n]]]) Exp[Exp[ProductLog[n]] - 1])/ProductLog[n]^n /; (n -> 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> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BellB] </annotation> </semantics> <mi> n </mi> </msub> <mo> &#8733; </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> W </mi> <annotation encoding='Mathematica'> TagBox[&quot;W&quot;, ProductLog] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <semantics> <mi> W </mi> <annotation encoding='Mathematica'> TagBox[&quot;W&quot;, ProductLog] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </msup> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <msup> <mi> &#8519; </mi> <mrow> <semantics> <mi> W </mi> <annotation encoding='Mathematica'> TagBox[&quot;W&quot;, ProductLog] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> W </mi> <annotation encoding='Mathematica'> TagBox[&quot;W&quot;, ProductLog] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </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> BellB </ci> <ci> n </ci> </apply> <apply> <times /> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <apply> <power /> <apply> <ci> ProductLog </ci> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <ci> ProductLog </ci> <ci> n </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <power /> <exponentiale /> <apply> <ci> ProductLog </ci> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> ProductLog </ci> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> n </ci> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02