\sum_{m=0}^nmD_n(m) = \sum_{m=0}^n m \left( \begin{array}{c} n \\ m \end{array} \right) D_{n-m} = n\sum_{m=0}^n \frac{(n-1)!}{(m-1)!(n-m)!}D_{n-m} = \\ = n \left [ \underline {\sum_{m=0}^n \left( \begin{array}{c} n-1 \\ m-1 \end{array} \right)D_{n-m}} \right] = ~ ...~