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!
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).