# Fungrim entry: 4d1365

${z \choose k} = \sum_{i=0}^{k} s\!\left(k, i\right) \frac{{z}^{i}}{k !}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}_{\ge 0}$
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${n \choose k}$ Binomial coefficient
Sum$\sum_{n} f(n)$ Sum
StirlingS1$s\!\left(n, k\right)$ Signed Stirling number of the first kind
Pow${a}^{b}$ Power
Factorial$n !$ Factorial
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ 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))), For(i, 0, k)))),
Variables(z, k),
Assumptions(And(Element(z, CC), Element(k, ZZGreaterEqual(0)))))

## Topics using this entry

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

2021-03-15 19:12:00.328586 UTC