For a fixed finite group G, the numbers Ng of equivalence classes of orientation-preserving actions of G on closed orientable surfaces Σg of genus g can be encoded by a generating function [sum ]Ngzg. When equivalence is determined by the isomorphism class of the quotient orbifold Σg/G, we show that the generating function is rational. When equivalence is topological conjugacy, we examine the cases where G is abelian and show that the generating function is again rational in the cases where G is cyclic.