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 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)."
is a formal power series (note that the lower bound of summation is 1, not 0), and
with an and bn as above, then