BellB[n] == Ceiling[(1/E) Sum[k^n/k!, {k, 1, 2 n}]] /; Element[n, Integers] && n > 0

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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> &#63449; </mo> <mrow> <mo> &#8968; </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> &#8519; </mi> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </munderover> <mfrac> <msup> <mi> k </mi> <mi> n </mi> </msup> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> </mrow> <mo> &#8969; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> BellB </ci> <ci> n </ci> </apply> <apply> <ceiling /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <exponentiale /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </uplimit> <apply> <times /> <apply> <power /> <ci> k </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BellB", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Ceiling", "[", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["2", " ", "n"]]], FractionBox[SuperscriptBox["k", "n"], RowBox[List["k", "!"]]]]], "\[ExponentialE]"], "]"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

2007-05-02