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Write the generating function for the number of partitions of $n$ into parts $1, 2, . . . , m$ each of which is used an odd number of times.
I've gotten myself confused, could someone give me a hint please? I have $\prod_{k\geq 1} \frac{1+x}{1-x^k}$, but I'm unsure.
Assuming all of the elements in $[m]$ must be used. Then the coefficient of $x^n$ in \begin{eqnarray*} \prod_{i=1}^{m} \frac{x^i y}{(1-x^{2i} y^2)} \end{eqnarray*} will be a polynomial in $y$. This polynomial will grade the number of parts in each partition. If you set $y=1$ will get the number of partitions of $n$ into an odd number of parts using all the elements of $[m]$.
