<s>[INST] Generating functions (Expansion) Problem: Give an interpretation in partition terms for the coefficient $x^{12}$ in the expansion $(1+x^2+x^4+x^6+x^8+x^{10}+x^{12})(1+x^4+x^8+x^{12})(1+x^6+x^{12})(1+x^8)(1+x^{10})(1+x^{12})$
I'm not sure if I have to find a generating function or only determine when $x^{12}$ appears or only try to describe his behavior.
Thanks for your help.
 [/INST] It's the same as the coefficient of $x^{12}$ in the product
$$\eqalign{
  &(1+x^2+x^4+\cdots)(1+x^4+x^8+\cdots)(1+x^6+x^{12}+\cdots)\cr
  &\quad{}\times(1+x^8+x^{16}+\cdots)(1+x^{10}+x^{20}+\cdots)\cr
  &\quad{}\times(1+x^{12}+x^{24}+\cdots)(1+x^{14}+x^{28}+\cdots)\cdots\ ,\cr}$$
because the extra terms I have included cannot possibly be part of any product giving $x^{12}$.  This in turn can be written as
$$(1+x^2+x^{2+2}+\cdots)(1+x^4+x^{4+4}+\cdots)(1+x^6+x^{6+6}+\cdots)\cdots\ ,$$
and the coefficient of $x^{12}$ will be the number of ways you can write $12$ as the sum of zero or more $2$s, zero or more $4$s and so on.  In other words, it is

the number of partitions of $12$ into even summands.

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