# Stirling transform

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

$b_n=\sum_{k=1}^n \left\{\begin{matrix} n \\ k \end{matrix} \right\} a_k,$

where $\left\{\begin{matrix} n \\ k \end{matrix} \right\}$ 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

$a_n=\sum_{k=1}^n s(n,k) b_k,$

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

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

If

$f(x)=\sum_{n=1}^\infty {a_n \over n!} x^n$

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

$g(x)=\sum_{n=1}^\infty {b_n \over n!} x^n$

with an and bn as above, then

$g(x)=f(e^x-1).\,$