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 where
is the Stirling number of the second kind, which is the number of partitions of a set of size
This is a linear sequence transformation.
is a signed Stirling number of the first kind, where the unsigned
can be defined as the number of permutations on
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 is a formal power series, and with an and bn as above, then Likewise, the inverse transform leads to the generating function identity