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, 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 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 See also Binomial transform Generating function transformation List of factorial and binomial topics References . Khristo N. Boyadzhiev, Notes on the Binomial Transform, Theory and Table, with Appendix on the Stirling Transform (2018), World Scientific. Factorial and binomial topics Transforms