Prove the following identity by counting two sets of partitions in two different ways.
\begin{equation}\prod_{i\geq 0}(1+x^{2i+1})=1 + \sum_{n\gt 0}x^{n^2}\prod_{j=1}^n\frac{1}{1-x^{2j}}.\end{equation}
I know the partition on the left side, it's the number of partitions with $i$ distinct odd parts. I just need help doing the right side. The second part of the second term , the product is the number of partitions of $n$ and for each $i$, the parts of size $i$ occur an even number of times. Now it's being multiplied by the summations of $x^{n^2}$ times. How would that effect the partition?