Power series for an infinite product.

Question

Let $x$ be a real number with magnitude less than 1. Develop in a power series in $z$ the infinite product:

$$(1+xz)(1+x^{2}z)(1+x^{3}z)\cdots$$

I have tried multiplying terms in various ways, but have yet to find a pattern. Any suggestions?

Answer 1

The coefficient of $z^k$ in the product $$ (1+xz)(1+x^2 z)(1+x^3 z)\cdot\ldots \tag{1}$$ is given by the sum $$ \sum_{\substack{\,\,A\subset \mathbb{N}^+ \\ |A|=k}}\prod_{a\in A}x^a =\sum_{m\geq \frac{k(k+1)}{2}}Q(m,k)\,x^m\tag{2}$$ where $Q(m,k)$ denotes the ways for partitioning $m$ into $k$ distinct parts.
$(1)$ multiplied by $(1+z)$ is a q-Pochhammer symbol.