\[\sum_{n=0}^{\infty}p_{k}\left(n\right)q^{n}=\prod_{j=1}^{k}\frac{1}{1-q^{j}}=1%
+\sum_{m=1}^{\infty}\genfrac{[}{]}{0.0pt}{}{k+m-1}{m}_{q}q^{m},\]