Fungrim entry: 6189b9

\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:

n \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`	$\left\{{n \atop k}\right\}$	Stirling number of the second kind
`Factorial`	$n !$	Factorial
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Binomial`	${n \choose k}$	Binomial coefficient
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	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)))))

Topics using this entry

Stirling numbers