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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?

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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 $x$.

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