# Distinct partitions of summands using generating function.

Consider the sequence $a_0, a_1, a_2, . . . ,$ where $a_n$ is the number of partitions of n into distinct even summands. Find the generating function for the sequence.

i know that $\prod_{i=1}^\infty (1+x^i)$ is the formula for distinct summands being even seems to imply that i want $g(x)= \prod_{i=1}^\infty (1+x^{2i})$ but this simply doesn't define the the odd terms (like 7) can i assume the coefficient of all the odd powers of x is 0?

Yes, all the coefficients of odd powers in $\prod_{i=1}^\infty (1+x^{2i})$ will be zero. Think about multiplying out the product
$$(1+x^2)(1+x^4)(1+x^6) \cdots$$
From each factor you either pick $1$ (which is $x^0)$ or some even power of $x$. Multiplying these together gives an even power of $x$.