Subfactorial[n] == Floor[(n! + 1)/E] /; 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> <mrow> <mi> Subfactorial </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> &#8519; </mi> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[List[], Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Subfactorial </ci> <ci> n </ci> </apply> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <exponentiale /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Subfactorial", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["n", "!"]], "+", "1"]], "\[ExponentialE]"], "]"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

2007-05-02