Let $\displaystyle p_{\leq c}(n)$ represent the number of partitions of $n$ into at most $c$ parts. What is the generating function of $\displaystyle\sum_{n \geq 0} p_{\leq c}(n)x^n$?
I'm completely stumped by this problem, not even sure where to start. I know that if the coefficients were just the number of partitions of $n$ without restriction, then the generating function would be $\displaystyle\prod_{n=1}^\infty \dfrac{1}{1-x^n}$, but I don't know how to use this fact. I also suspect there's a relationship between $p_{\leq c}(n)$ and $p_{\leq c}(n-1)$, $p_{\leq c}(n-2)$, and so on that I can use, but I have no idea what.
Thanks!