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In combinatorics, the nth Bell number is the number of partitions of a set of size n.  All of the cumulants of the sequence of Bell numbers are equal to 1.  The Bell numbers are the moments of the Poisson distribution with nilai ekspektasi 1.

Cumulants of a polynomial sequence of binomial type

For any sequence { κn : n = 1, 2, 3, ... } of scalars in a field of characteristic zero, being considered formal cumulants, there is a corresponding sequence { μ ′ : n = 1, 2, 3, ... } of formal moments, given by the polynomials above.  For those polynomials, construct a polynomial sequence in the following way.  Out the polynomial

make a new polynomial in these plus one additional variable x:

... and generalize the pattern.  The pattern is that the numbers of blocks in the aforementioned partitions are the exponents on x.  éach coefficient is a polynomial in the cumulants; these are the Bell polynomials, named after Eric Temple Bell.

This sequence of polynomials is of binomial type.  In fact, no other sequences of binomial type exist; every polynomial sequence of binomial type is completely determined by its sequence of formal cumulants.