Det[Table[BellB[k + l], {k, 0, n}, {l, 0, n}]] == Product[k!, {k, 1, n}]

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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> <semantics> <mrow> <mo> &#10072; </mo> <msub> <mrow> <mo> ( </mo> <msub> <mi> B </mi> <mrow> <mi> k </mi> <mo> + </mo> <mi> l </mi> </mrow> </msub> <mo> ) </mo> </mrow> <mtable> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> &#8804; </mo> <mi> k </mi> <mo> &#8804; </mo> <mi> n </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> &#8804; </mo> <mi> l </mi> <mo> &#8804; </mo> <mi> n </mi> </mrow> </mtd> </mtr> </mtable> </msub> <mo> &#10072; </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[LeftBracketingBar]&quot;, SubscriptBox[RowBox[List[&quot;(&quot;, SubscriptBox[TagBox[&quot;B&quot;, BellB], RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;l&quot;]]], &quot;)&quot;]], GridBox[List[List[RowBox[List[&quot;0&quot;, &quot;\[LessEqual]&quot;, &quot;k&quot;, &quot;\[LessEqual]&quot;, &quot;n&quot;]]], List[RowBox[List[&quot;0&quot;, &quot;\[LessEqual]&quot;, &quot;l&quot;, &quot;\[LessEqual]&quot;, &quot;n&quot;]]]]]], &quot;\[RightBracketingBar]&quot;]], List[Det]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <determinant /> <apply> <ci> Subscript </ci> <apply> <ci> BellB </ci> <apply> <plus /> <ci> k </ci> <ci> l </ci> </apply> </apply> <list> <list> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> k </ci> <ci> n </ci> </apply> </list> <list> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> l </ci> <ci> n </ci> </apply> </list> </list> </apply> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <factorial /> <ci> k </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02