Stirling transform

In combinatorial mathematics, the Stirling transform of a sequence { a_n : n = 1, 2, 3, ... } of numbers is the sequence { b_n : n = 1, 2, 3, ... } given by

where { n k } is the Stirling number of the second kind, also denoted S(n,k) (with a capital S), which is the number of partitions of a set of size n into k parts.

The inverse transform is

where s(n,k) (with a lower-case s) is a Stirling number of the first kind.

Berstein and Sloane (cited below) state "If a_n is the number of objects in some class with points labeled 1, 2, ..., n (with all labels distinct, i.e. ordinary labeled structures), then b_n is the number of objects with points labeled 1, 2, ..., n (with repetitions allowed)."

If

is a formal power series (note that the lower bound of summation is 1, not 0), and

with a_n and b_n as above, then