A Dobiński series for the number of surjective mappings (Q670727)

It is well known that the number of surjective mappings of an $r$-element set onto a $j$-element set is given by $j!\begin{Bmatrix}r\\j\end{Bmatrix}$, where $\begin{Bmatrix}r\\j\end{Bmatrix}$ denotes a Stirling number of the second kind. Let $S_r:=\sum_{j=1}^r j! \begin{Bmatrix}r\\j\end{Bmatrix}$. In this present paper, the author shows that $S(r)=\frac{1}{2}\sum_{k=1}^\infty\frac{k^r}{2^k}$, which may be considered as an analog of the Dobiski series for the $r$th Bell number. The proof is based on a combinatorial argument.