I need help finding the generating function for $R_n$, where $R_n$ is the number of partitions of $n$ in which each part occurs at most $3$ times...
I know that it is an infinite product and might look like this
$$R_n = (1 +x + x^2 + x^3)\cdot(1 + x^2 + x^4 + x^6)\cdot \ldots$$ which might simplify to..
$$R_n = \frac{1}{1-x^3}$$ but I'm not too sure.
The ordinary generating function for $R_n$ is $$\sum_{n=0}^\infty R_nx^n=(1+x+x^2+x^3)(1+x^2+x^4+x^6)(1+x^3+x^6+x^9)(1+x^4+x^8+x^{12})\cdot\cdots=\prod_{n=1}^\infty(1+x^n+x^{2n}+x^{3n})=\prod_{n=1}^\infty\frac{1-x^{4n}}{1-x^n}.$$