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Fungrim entry: 4d1365

(zk)=∑i=0ks ⁣(k,i)zik!{z \choose k} = \sum_{i=0}^{k} s\!\left(k, i\right) \frac{{z}^{i}}{k !}
Assumptions:z∈C and⁡ k∈Z≥0z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}
TeX:
{z \choose k} = \sum_{i=0}^{k} s\!\left(k, i\right) \frac{{z}^{i}}{k !}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Binomial(nk){n \choose k} Binomial coefficient
StirlingS1s ⁣(n,k)s\!\left(n, k\right) Signed Stirling number of the first kind
Powab{a}^{b} Power
Factorialn!n ! Factorial
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZ≥n\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("4d1365"),
    Formula(Equal(Binomial(z, k), Sum(Mul(StirlingS1(k, i), Div(Pow(z, i), Factorial(k))), Tuple(i, 0, k)))),
    Variables(z, k),
    Assumptions(And(Element(z, CC), 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.

2019-06-18 07:49:59.356594 UTC