Compute $\frac{1}{e}\sum\limits_{n=0}^{\infty}\frac{n^{k}}{n!}$ for $k=0, 1, 2 ... $
$$\Large\sum_{n=1}^{\infty}\frac{\mathbb e^{nx}}{n!}=\mathbb e^{\mathbb e^x}-1$$
$$~$$
$$\Large\sum_{n=1}^\infty \frac{n^k}{n!}=\frac{d^k}{dx^k}(\mathbb e^{\mathbb e^x}-1)\huge]_{x=0}$$