Sequences, Series, and the Binomial Theorem · Superchallenge problem

Find the exact value of $\displaystyle\sum_{k=1}^{\infty}\frac{k\cdot 2^k}{(k+2)!}=\frac{2}{3!}+\frac{2\cdot 4}{4!}+\frac{3\cdot 8}{5!}+\cdots$.