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What is a generating function for the sequence $\{a_n\}_{n\ge1}$ where $a_n$ is the number of partitions of $n$ with only distinct even parts, and how would you show that it was found? The only formulas I've been able to find online involve distinct even parts and any number of odd parts, only distinct parts or only odd parts.

Thank you for any help you may be able to give me.

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$a_n$ is trivially zero for odd $n$ and a bijection between distinct even and distinct odd partitions exists for even $n$, so its generating function is simply that for distinct parts but substituting $x^2$ for $x$. Concretely this is a spaced-out OEIS A9: $$1+x^2+x^4+2x^6+2x^8+\dots$$

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