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r-Bell number

Bn,rB_{n, r} is the number of ways to partition an (n+r)(n+r) element set into blocks, such that the first rr elements are in different blocks.

Cases

Formula

Bn,r=∑k=0n(∑i=0n(ni)(ik)rn−i)B_{n,r} = \sum_{k=0}^n \left( \sum_{i=0}^n \binom{n}{i} \binom{i}{k} r^{n-i} \right)Bn,r=1e∑k=0∞(k+r)nk!B_{n,r} = \frac{1}{e} \sum_{k=0}^\infty \frac{(k + r)^n}{k!}Bn,r=2n!πe Im∫0πeeiθereiθsin⁡(nθ) dθB_{n,r} = \frac{2n!}{\pi e} \, \mathrm{Im} \int_{0}^\pi e^{e^{i\theta}} e^{r e^{i\theta}} \sin(n\theta) \, d\theta

Recurrence

Bn,r=Bn−1,r+1+rBn−1,rB_{n,r} = B_{n-1,r+1} + r B_{n-1,r}Bn,r=rBn−1,r+∑k=0n−1(n−1k)Bk,rB_{n,r} = r B_{n-1,r} + \sum_{k=0}^{n-1} \binom{n-1}{k} B_{k,r}Bn,r+1=∑k=0n(nk)Bk,rB_{n,r+1} = \sum_{k=0}^n \binom{n}{k} B_{k,r}Bn,r=∑k=0n(nk)(−1)n−kBk,r+1B_{n,r} = \sum_{k=0}^n \binom{n}{k} (-1)^{n-k} B_{k,r+1}

Generating functions

∑n=0∞Bn,rznn!=eez−1+rz\sum_{n=0}^\infty \frac{B_{n,r} z^n}{n!} = e^{e^z-1 + rz}

References

Mező: The r-Bell Numbers

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