Are you familiar with generating functions?

If so...

Let $\displaystyle o_n,e_n$ be the number of ways to partition $\displaystyle n$ into at most $\displaystyle n$ odd parts and the number of partitions of $\displaystyle n$ such that each even part occurs an even number of times, respectively. Define $\displaystyle O(x),E(x)$ to be the respective generating functions for $\displaystyle o_n,e_n$, then

$\displaystyle O(x)=\sum_{n\ge0}o_nx^n=\prod_{r\,\text{odd}}\left (\sum_{k=0}^nx^{rk}\right)=\prod_{r\,\text{odd}}(1 +x^r+\cdots+x^{rn})$

and

$\displaystyle E(x)=\sum_{n\ge0}e_nx^n=\prod_{r\,\text{even}}\lef t(\sum_{k\ge0}x^{rk}\right)=\prod_{r\,\text{even}} \frac{1}{1-x^{2r}}$

Let $\displaystyle P(x)=O(x)E(x)$, then

$\displaystyle P(x)=\prod_{r\,\text{odd}}(1+x^r+\cdots+x^{rn})\pr od_{r\,\text{even}}\frac{1}{1-x^{2r}}=\sum_{n\ge 0} p_nx^n,$

where $\displaystyle p_n$ is the number of partitions of $\displaystyle n$ where the number of even parts is even. The number of ways to do this when $\displaystyle n=2k+1$ is the coefficient of $\displaystyle x^{2k+1}$.

If not...I highly recommend

generatingfunctionology by Wilf because it's a good book, but better yet, it's free to download at the link I provided. In particular, page 100 begins his discussion of using generating functions for partitions.

Hope this helps.