From a Miklos Bona combinatorics textbook,
I'm at an almost total loss. My professor recently discussed products of generating functions, so I suspected this problem might relate. The only strategy I came up with was to find a g.f. for $p_{e} (n)$, that is, the number of partitions of $n$ into even parts that are at most 6, and a separate one, say $p_{o} (n)$, or the number of ways of partitions of $n$ into at most one odd part, and multiply them.
For $p_{e} (n)$, I obtained $$p_{e} (n)=1+x^{2}+2x^{4}+3x^{6}+...$$ ... which seems fishy. For $p_{o} (n)$, I obtained $$p_{o} (n)=x+x^{3}+x^{5}+...$$... which again I'm not sure if that's quite right. I truly want to learn to do these types of problems, so any help just getting started would be much appreciated. Also, sorry for not posting more preliminary work - this thing has me on the ropes.
The GF for partitions into $2$s, $4$s and $6$s is $$f(x)=\frac1{(1-x^2)(1-x^4)(1-x^6)}.$$ The GF for partitions into exactly one odd part is $$g(x)=x+x^3+x^5+\cdots=\frac x{1-x^2}.$$ The GF for partitions into exactly one odd part and other parts in $\{2,4,6\}$ is $f(x)g(x)$ etc.