Bell number

The Bell's number, number or Bell exponential number is the number of partitions of an n- element set. It is named after the mathematician Eric Temple Bell. The sequence B0, B1, B2, B3, ... starting with

Properties

For Bell's numbers, the recursive formula are

And the formula ( Dobiński 1877)

Thus the -th moment of a Poisson distribution with the expected value is 1

The generating function of the Bell numbers is

The exponential generating function is

In addition, the Bell numbers satisfy the congruence ( Touchard 1933)

For natural numbers and prime numbers, and in particular and, by iteration,

It is believed that the smallest period is. For primes

For the congruence applies.

Since the Stirling number of the second kind, the number of partitions is one - element set, applies

Asymptotics

For the Bell numbers different asymptotic formulas are known about

With the Lambert -W function.