Determine the coefficient of $~x ^ {12}~$ in:
$(1+^2+^4+^6+^8+^{10}+^{12})(1+^4+^8+^{12})(1+^6+^{12})(1+^8)(1+^{10})(1+^{12})$
How to proceed with the resolution of this type of question when there is the product of more than two functions?
2026-03-29 06:27:16.1774765636
Extraction of coefficient from Generating Function
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The coefficient of $x^{12}$ is equal to the number of partitions of $12$ in which all summands are even.
Given a partition of $12$ in which all summands are even we can divide each summand by $2$ to get a partition of $6$. And given a partition of $6$ we can find a partition of $12$ with even summands by doubling each summand. So there is a one-to-one correspondence between the partitions of $12$ in which all summands are even and the partitions of $6$.
Therefore the coefficient of $x^{12}$ is equal to the number of partitions of $6$.