The procedure bell computes the 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 th Bell number has several interesting interpretations, including

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