# §26.7 Set Partitions: Bell Numbers

## §26.7(i) Definitions

$\mathop{B\/}\nolimits\!\left(n\right)$ is the number of partitions of $\{1,2,\ldots,n\}$. For $\mathop{S\/}\nolimits\!\left(n,k\right)$ see §26.8(i).

 26.7.1 $\mathop{B\/}\nolimits\!\left(0\right)=1,$ Symbols: $\mathop{B\/}\nolimits\!\left(\NVar{n}\right)$: Bell number Permalink: http://dlmf.nist.gov/26.7.E1 Encodings: TeX, pMML, png See also: Annotations for 26.7(i)
 26.7.2 $\mathop{B\/}\nolimits\!\left(n\right)=\sum_{k=0}^{n}\mathop{S\/}\nolimits\!% \left(n,k\right),$
 26.7.3 $\mathop{B\/}\nolimits\!\left(n\right)=\sum_{k=1}^{m}\frac{k^{n}}{k!}\sum_{j=0}% ^{m-k}\frac{(-1)^{j}}{j!},$ $m\geq n$,
 26.7.4 $\mathop{B\/}\nolimits\!\left(n\right)={\mathrm{e}^{-1}}\sum_{k=1}^{\infty}% \frac{k^{n}}{k!}=1+\left\lfloor{\mathrm{e}^{-1}}\sum_{k=1}^{2n}\frac{k^{n}}{k!% }\right\rfloor.$

See Table 26.7.1.

## §26.7(ii) Generating Function

 26.7.5 $\sum_{n=0}^{\infty}\mathop{B\/}\nolimits\!\left(n\right)\frac{x^{n}}{n!}=% \mathop{\exp\/}\nolimits({\mathrm{e}^{x}}-1).$

## §26.7(iii) Recurrence Relation

26.7.6 $\mathop{B\/}\nolimits\!\left(n+1\right)=\sum_{k=0}^{n}\binom{n}{k}\mathop{B\/}% \nolimits\!\left(k\right).$

## §26.7(iv) Asymptotic Approximation

 26.7.7 $\mathop{B\/}\nolimits\!\left(n\right)=\frac{N^{n}{\mathrm{e}^{N-n-1}}}{(1+% \mathop{\ln\/}\nolimits N)^{1/2}}\left(1+\mathop{O\/}\nolimits\left(\frac{(% \mathop{\ln\/}\nolimits n)^{1/2}}{n^{1/2}}\right)\right),$ $n\to\infty$,

where

 26.7.8 $N\mathop{\ln\/}\nolimits N=n,$ Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $n$: nonnegative integer and $N$ Permalink: http://dlmf.nist.gov/26.7.E8 Encodings: TeX, pMML, png See also: Annotations for 26.7(iv)

or, equivalently, $N={\mathrm{e}^{\mathop{\mathrm{Wm}\/}\nolimits\!\left(n\right)}}$, with properties of the Lambert function $\mathop{\mathrm{Wm}\/}\nolimits\!\left(n\right)$ given in §4.13. For higher approximations to $\mathop{B\/}\nolimits\!\left(n\right)$ as $n\to\infty$ see de Bruijn (1961, pp. 104–108).