Are you familiar with generating functions?

If so...

Let

be the number of ways to partition

into at most

odd parts and the number of partitions of

such that each even part occurs an even number of times, respectively. Define

to be the respective generating functions for

, then

and

Let

, then

where

is the number of partitions of

where the number of even parts is even. The number of ways to do this when

is the coefficient of

.

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.