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r-Bell Numbers

Bn,rB_{n, r}

is the number of ways of partitioning the set {1,2,...,n+r}\{1, 2, ..., n + r\} such that {1,2,...,r}\{1, 2, ..., r\} are in different blocks.

They are the sum of the r-Stirling numbers of the second kind, but take care about notation, as some places consider partitions of the set {1, 2, ..., n + r} (like this page) and others of the set {1, 2, ..., n} (like the stirling page).

Bn,r=k=0n{n+rk+r}rB_{n,r} = \sum_{k=0}^{n} \left\{ \begin{matrix} n + r \\ k + r \end{matrix} \right\}_r

Bn=Bn,0B_n = B_{n, 0}

The main source of this page is the article linked below. A lot of the theorems were given for r-Bell polynomials, and I tried to translate them to r-Bell numbers. In case some formula here is giving wrong results, please see the original!


Bn,r=0 for n<rB_{n, r} = 0 \text{ for } n < r

Bn,r=1 for n=rB_{n, r} = 1 \text{ for } n = r

Bn,r=rBn1,r+k=0n1(n1k)Bn1k,rB_{n,r} = r B_{n-1,r} + \sum_{k=0}^{n-1} \binom{n-1}{k} B_{n-1-k,r}


Bn,r=rBn1,r+Bn1,r+1B_{n,r} = r B_{n-1,r} + B_{n-1,r+1}


Bn,r=1ek=0(k+r)nk!B_{n,r} = \frac{1}{e} \sum_{k=0}^{\infty} \frac{(k + r)^n}{k!}

Bn,r=k=0n(i=0n(ni){ik}rni)B_{n,r} = \sum_{k=0}^{n} \left( \sum_{i=0}^{n} \binom{n}{i} \left\{ \begin{matrix} i \\ k \end{matrix} \right\} r^{n-i} \right)

Bn,r=2n!πe0πeeiθereiθsin(nθ)dθ.B_{n,r} = \frac{2n!}{\pi e} \Im \int_{0}^{\pi} e^{e^{i\theta}} e^{re^{i\theta}} \sin(n\theta) \, d\theta.


Bn+m,r=j=0m{m+rj+r}rBn,r+j.B_{n+m,r} = \sum_{j=0}^{m} \left\{ \begin{matrix} m + r \\ j + r \end{matrix} \right\}_r B_{n,r+j}.

Bn,r+m=j=0m(1)mj[m+rj+r]rBn+j,r.B_{n,r+m} = \sum_{j=0}^{m} (-1)^{m-j} \left[ \begin{matrix} m + r \\ j + r \end{matrix} \right]_r B_{n+j,r}.

This one might be wrong, as it refers to the polynomials and I tried to adapt it to just the numbers. Check the article just in case.

Bn,r=k=0nrk(nk)BnkB_{n,r} = \sum_{k=0}^{n} r^k \binom{n}{k} B_{n-k}

Generating functions

Exponential generating function

n=0Bn,rznn!=e(ez1)+rz\sum_{n=0}^{\infty} B_{n,r} \frac{z^n}{n!} = e^{(e^z-1)+rz}


Istvan Mezo: The r-Bell Numbers