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 formula_2 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, and
with "a""n" and "b""n" as above, then
Likewise, the inverse transform leads to the generating function identity 