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Here is an observation (thanks to OEIS):

$$\sum_{i=0}^\infty \frac{i^k}{i!}= B_k e,$$ where $B_k$ is the $k$-th Bell number. I might be having reading comprehension issues, but I don't see this formula in the OEIS notes. I assume this is very well known - can someone point me at a reference or a simple proof?

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That is essentially Dobinski's formula.

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  • $\begingroup$ Cool! I am surprised OEIS does not mention it... $\endgroup$
    – Igor Rivin
    Sep 28, 2015 at 10:36
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    $\begingroup$ This mentioned a couple of times at least in the notes on A00110, by Jonathan Love in 2007 and by Wolfdieter Lang in 2015. It is in the formulas section by Benoit Cloitre in 2002. $\endgroup$ Sep 28, 2015 at 10:51
  • $\begingroup$ It's more than "essentially"---it is almost precisely the linked formula :-) $\endgroup$
    – Suvrit
    Sep 28, 2015 at 15:39
  • $\begingroup$ And a generlized Dobinski formula is used in the proof of a formula for the o.g.f. of the Bell numbers cited in the related questions! $\endgroup$ Sep 29, 2015 at 2:54
  • $\begingroup$ I didn't know it was Dobinski's formula, but this identity is Exercise 1.9(a) in the first edition of Lovász's Combinatorial Problems and Exercises. $\endgroup$
    – bof
    Feb 26, 2016 at 2:33

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