BellB[n] == ((2 n!)/(Pi E)) Integrate[E^(E^Cos[t] Cos[Sin[t]]) Sin[E^Cos[t] Sin[Sin[t]] Sin[n t]], {t, 0, Pi}]

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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> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BellB] </annotation> </semantics> <mi> n </mi> </msub> <mo> &#63449; </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8519; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> &#960; </mi> </msubsup> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> BellB </ci> <ci> n </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <apply> <times /> <pi /> <exponentiale /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <pi /> </uplimit> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <cos /> <ci> t </ci> </apply> </apply> <apply> <cos /> <apply> <sin /> <ci> t </ci> </apply> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <cos /> <ci> t </ci> </apply> </apply> <apply> <sin /> <apply> <sin /> <ci> t </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> n </ci> <ci> t </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02