\[\sum_{n=0}^{\infty}p\left(\mathcal{D}k,n\right)q^{n}={1+\sum_{m=1}^{\infty}%
\frac{q^{(km^{2}+(2-k)m)/2}}{(1-q)(1-q^{2})\cdots(1-q^{m})}},\]