Generating function for division of $n$ into smaller subsets. I need to find the generating function for the number of ways of dividing $n$ into parts out of even numbers. The ways are only different if their parts are different, meaning that $2+1+2$ and $2+2+1$ are the same, because only their arrangmenent is different.
If we would like all of them to be distinct, then we would have $F(x)=(1+x^2)(1+x^4)(1+x^6)...$, but what if some can be repeated?  Is it $\prod\limits_{n=0}^{\infty}(1+x^2+x^4+x^6...)$?

If the parts can be repeated, this generating function can be written as
\begin{align}
F(x) & = (1 + x^{2} + x^{2+2} + \ldots)\cdot (1 + x^{4}+x^{4+4}+\ldots)\cdot (1 + x^{6}+x^{6+6}+\ldots)\cdot \ldots\\[0.7em]
& = \dfrac{1}{1-x^{2}}\cdot \dfrac{1}{1-x^{4}} \cdot \dfrac{1}{1-x^{6}} \cdot \ldots\\[0.9em]
& = \prod_{n=1}^{\infty}\dfrac{1}{1-x^{2n}}
\end{align}
