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Stirling numbers

Table of contents: Tables - Recurrence relations - Connection formulas - Generating functions - Sum representations - Sums - Bounds and inequalities

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Symbol: StirlingCycle [nk]\left[{n \atop k}\right] Unsigned Stirling number of the first kind
2e9d0c
Symbol: StirlingS1 s ⁣(n,k)s\!\left(n, k\right) Signed Stirling number of the first kind
4c6c43
Symbol: StirlingS2 {nk}\left\{{n \atop k}\right\} Stirling number of the second kind
1706bb
Symbol: BellNumber BnB_{n} Bell number

Tables

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Table of [nk]\left[{n \atop k}\right] for 0n100 \le n \le 10 and 0k100 \le k \le 10
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Table of s ⁣(n,k)s\!\left(n, k\right) for 0n100 \le n \le 10 and 0k100 \le k \le 10
cecede
Table of {nk}\left\{{n \atop k}\right\} for 0n100 \le n \le 10 and 0k100 \le k \le 10
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Table of BnB_{n} for 0n400 \le n \le 40

Recurrence relations

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[n+1k]=n[nk]+[nk1]\left[{n + 1 \atop k}\right] = n \left[{n \atop k}\right] + \left[{n \atop k - 1}\right]
18ec99
s ⁣(n+1,k)=s ⁣(n,k1)ns ⁣(n,k)s\!\left(n + 1, k\right) = s\!\left(n, k - 1\right) - n s\!\left(n, k\right)
9fbe4f
{n+1k}=k{nk}+{nk1}\left\{{n + 1 \atop k}\right\} = k \left\{{n \atop k}\right\} + \left\{{n \atop k - 1}\right\}

Connection formulas

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

Generating functions

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(x)n=k=0n[nk]xk\left(x\right)_{n} = \sum_{k=0}^{n} \left[{n \atop k}\right] {x}^{k}
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(xn+1)n=k=0ns ⁣(n,k)xk\left(x - n + 1\right)_{n} = \sum_{k=0}^{n} s\!\left(n, k\right) {x}^{k}
b823b0
xn=k=0n{nk}(xn+1)n{x}^{n} = \sum_{k=0}^{n} \left\{{n \atop k}\right\} \left(x - n + 1\right)_{n}
b01280
(log ⁣(1+x))kk!=n=k(1)nk[nk]xnn!\frac{{\left(\log\!\left(1 + x\right)\right)}^{k}}{k !} = \sum_{n=k}^{\infty} {\left(-1\right)}^{n - k} \left[{n \atop k}\right] \frac{{x}^{n}}{n !}
a9a610
(ex1)kk!=n=k{nk}xnn!\frac{{\left({e}^{x} - 1\right)}^{k}}{k !} = \sum_{n=k}^{\infty} \left\{{n \atop k}\right\} \frac{{x}^{n}}{n !}

Sum representations

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{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}

Sums

ea9e2f
k=0n[nk]=n!\sum_{k=0}^{n} \left[{n \atop k}\right] = n !
255576
k=0n{nk}=Bn\sum_{k=0}^{n} \left\{{n \atop k}\right\} = B_{n}

Bounds and inequalities

7774a3
[nk]2nn!k!\left[{n \atop k}\right] \le \frac{{2}^{n} n !}{k !}

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2019-05-23 08:00:13.607731 UTC