# Generating function for the number of partitions into parts 1,2,..,m, each of which is used an odd number of times

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 $$[m]$$.