Are there methods for obtaining the coefficient of $x^n$ in a generating function like $$\prod_{i=1}^\infty Q(x),$$ where $Q(x)$ is a rational function? This arises when we want to count partitions of integers.
I have searched in Wilf's generatingfunctionology, Goulden and Jackson's Combinatorial Enumeration, Riordan's Introduction to Combinatorics, and some books by George Andrews (his Number Theory book, and his two books on integer partitions), but can't find even an answer for very restricted cases.
Did I miss something, or is there really no known method for that?