The procedure bell computes the $n$th Bell number if the argument n is an integer; otherwise, it returns the unevaluated function call. For the BellB polynomials see BellB.

The Bell numbers are defined by the exponential generating function:

The Bell numbers are computed using the umbral definition :

where bell()^n represents bell(n).

For example:

The $n$th Bell number has several interesting interpretations, including

The command with(combinat,bell) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{combinat}\,\mathrm{bell}\right)$

$\left[\mathrm{bell}\right]$

$\mathrm{bell}\left(1\right)$

$1$

$\mathrm{bell}\left(4\right)$

$15$

$\mathrm{bell}\left(-1\right)$

$1$

$\mathrm{bell}\left(n\right)$

$\mathrm{bell}\left(n\right)$