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 ${\displaystyle \left\{{\begin{matrix}n\\k\end{matrix}}\right\}}$ is the Stirling number of the second kind, which is the number of partitions of a set of size ${\displaystyle n}$ into ${\displaystyle k}$ parts. This is a linear sequence transformation.

The inverse transform is

where ${\textstyle (-1)^{n-k}\left[{n \atop k}\right]}$ is a signed Stirling number of the first kind, where the unsigned ${\displaystyle \left[{n \atop k}\right]}$ can be defined as the number of permutations on ${\displaystyle n}$ elements with ${\displaystyle k}$ cycles.

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, and

with a_n and b_n as above, then