Stirling transform
In combinatorial mathematics, the Stirling transform of a sequence of numbers is the sequence given by
where is the Stirling number of the second kind, also denoted S, which is the number of partitions of a set of size n into k parts.
The inverse transform is
where s is a Stirling number of the first kind.
Berstein and Sloane state "If an is the number of objects in some class with points labeled 1, 2,..., n, then bn is the number of objects with points labeled 1, 2,..., n."
If
is a formal power series, and
with an and bn as above, then
Likewise, the inverse transform leads to the generating function identity
