# How to extract coefficient of $x^n$ in an infinite product generating function?

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?

• A first naïve guess would be that one must be lucky to find such a thing, since infinite products are rather nasty. However, I would also expect some nice bounds in some special cases and am interested to see those.
– HSN
Sep 18, 2013 at 15:55
• Did you really mean the factor in the product to be independent of $i$ (like taking an infinite power)? That would only makes sense if $Q(x)=1$. I suppose you meant $\prod_iQ_i(x)$ Sep 22, 2013 at 14:29

Such an infinite product is only defined in the sense of formal power series if all factors have constant term$~1$ (one might allow a finite number of them to have a different constant term, but this never happens in practice), and if the next smallest power of $x$ in $Q_i(x)$ becomes ever larger as $i$ increases. Then to compute the coefficient of a specific power $x^n$, one only needs to consider finitely many factors in the product that contribute to it. This is actually the reason that such infinite products are defined as formal power series.