Stirling numbers

Symbol: StirlingCycle $\left[{n \atop k}\right]$ Unsigned Stirling number of the first kind
Symbol: StirlingS1 $s\!\left(n, k\right)$ Signed Stirling number of the first kind
Symbol: StirlingS2 $\left\{{n \atop k}\right\}$ Stirling number of the second kind

Tables

Table of $\left[{n \atop k}\right]$ for $0 \le n \le 10$ and $0 \le k \le 10$
Table of $s\!\left(n, k\right)$ for $0 \le n \le 10$ and $0 \le k \le 10$
Table of $\left\{{n \atop k}\right\}$ for $0 \le n \le 10$ and $0 \le k \le 10$

Recurrence relations

$\left[{n + 1 \atop k}\right] = n \left[{n \atop k}\right] + \left[{n \atop k - 1}\right]$
$s\!\left(n + 1, k\right) = s\!\left(n, k - 1\right) - n s\!\left(n, k\right)$
$\left\{{n + 1 \atop k}\right\} = k \left\{{n \atop k}\right\} + \left\{{n \atop k - 1}\right\}$

Connection formulas

$s\!\left(n, k\right) = {\left(-1\right)}^{n + k} \left[{n \atop k}\right]$

Generating functions

$\left(x\right)_{n} = \sum_{k=0}^{n} \left[{n \atop k}\right] {x}^{k}$
$\left(x - n + 1\right)_{n} = \sum_{k=0}^{n} s\!\left(n, k\right) {x}^{k}$
${x}^{n} = \sum_{k=0}^{n} \left\{{n \atop k}\right\} \left(x - n + 1\right)_{n}$
$\frac{\log^{k}\!\left(1 + x\right)}{k !} = \sum_{n=k}^{\infty} {\left(-1\right)}^{n - k} \left[{n \atop k}\right] \frac{{x}^{n}}{n !}$
$\frac{{\left({e}^{x} - 1\right)}^{k}}{k !} = \sum_{n=k}^{\infty} \left\{{n \atop k}\right\} \frac{{x}^{n}}{n !}$

Sum representations

$\left\{{n \atop k}\right\} = \frac{1}{k !} \sum_{i=0}^{k} {\left(-1\right)}^{i} {k \choose i} {\left(k - i\right)}^{n}$

Sums

$\sum_{k=0}^{n} \left[{n \atop k}\right] = n !$
$\sum_{k=0}^{n} \left\{{n \atop k}\right\} = B_{n}$

Bounds and inequalities

$\left[{n \atop k}\right] \le \frac{{2}^{n} n !}{k !}$

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