A conjectured rational generating function

Question

In regard to my question here, let $G_n$ be a sequence of positive integers satisfying $\lim_{n\to\infty}G_n=\infty$, such that the generating function $\sum_{n\geq 1} G_nx^n$ is rational. Let $$ P_n(x) = \prod_{i=1}^n \left( 1+x^{G_i}\right), $$ or even more generally, $$ P_n(x) = \prod_{i=1}^n \left( 1+x^{G_i}+x^{2G_i}+\cdots + x^{rG_i}\right) $$ for some postive integer $r$ (independent from $n$). (Maybe even more general products are possible.) For a positive integer $k$, let $\nu_k(n)$ be the number of coefficients of the polynomial $P_n(x)$ that are equal to $k$. Is the generating function $F(x)=\sum_{n\geq 0}\nu_k(n)x^n$ rational? If so, what can be said about it? If $F(x)$ is not always rational, for what $G_n$ is it rational?