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Fungrim entry: 6189b9

{nk}=1k!i=0k(1)i(ki)(ki)n\left\{{n \atop k}\right\} = \frac{1}{k !} \sum_{i=0}^{k} {\left(-1\right)}^{i} {k \choose i} {\left(k - i\right)}^{n}
Assumptions:nZ0andkZ0n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}
TeX:
\left\{{n \atop k}\right\} = \frac{1}{k !} \sum_{i=0}^{k} {\left(-1\right)}^{i} {k \choose i} {\left(k - i\right)}^{n}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
StirlingS2{nk}\left\{{n \atop k}\right\} Stirling number of the second kind
Factorialn!n ! Factorial
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
Binomial(nk){n \choose k} Binomial coefficient
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("6189b9"),
    Formula(Equal(StirlingS2(n, k), Mul(Div(1, Factorial(k)), Sum(Mul(Mul(Pow(-1, i), Binomial(k, i)), Pow(Sub(k, i), n)), For(i, 0, k))))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(k, ZZGreaterEqual(0)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-11-11 15:50:15.016492 UTC