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Is there a formula for $\displaystyle\sum_{n=1}^{\infty}\frac{n^k}{n!}$ in terms of $k$?

What about $\displaystyle\sum_{n=1}^{\infty}\frac{(n-1)^k}{n!}$?

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  • $\begingroup$ I believe you have checked convergence.... $\endgroup$ – user87543 Dec 21 '13 at 13:03
  • $\begingroup$ Yes, it converges.. $\endgroup$ – user85798 Dec 21 '13 at 13:19
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They are of the form $B_ke$, where $B_k$ is a Bell number. They represent the number of partitions of a set with n members, as well as the coefficients of the Taylor series for $e^{e^x}$. See Dobinski's formula, whose proof can be found here. Their sequence is on OEIS, and their values can be computed using a formula similar to that of Pascal's triangle. As for your second sum, it is of the form $N_ke-$$(-1)^k$ where $N_k$ represent the number of partitions of a set of n elements into blocks of size greater than $1$ Their sequence, along with additional explanations as to their other meanings, can also be found on OEIS.

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