TITLE: Combinatorial interpretation of a generating function (infinite product)
QUESTION [0 upvotes]: What is the combinatorial interpretation of the coefficient of the $n^{th}$ term in the following infinite product
$\Pi_{k\geq 1}(1+z^k)$
I know it should somehow related to the number of partition, but I don't know exactly what it is. Also the infinite product just confuses me, since there is no bound on the number it is counting. 
Any help would be appreciated!

REPLY [2 votes]: The coefficient of $z^n$ in $\Pi_{k\geq 1}(1+z^k)$ is the number of ways in which we can write $n$ as a sum of distinct positive integers. See A000009 
Since
$$\prod_{k\geq 1}(1+z^k)=\frac{\prod_{k\geq 1}(1-z^{2k})}{\prod_{k\geq 1}(1-z^{k})}=\frac{1}{\prod_{k\geq 1}(1-z^{2k-1})}$$
it is also the number of ways in which we can write $n$ as a sum of odd positive integers (not necessarily distinct).
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