[ \sum_{n=0}^{\infty} {\frac{n^4}{n!}}=15e ]
This is strange enough to provoke wonder, but simple enough to serve as an entry-point to an interesting generalization.
where (B_n) is the nth Bell number. Dobiński published a paper about it in 1877, but it seems that he only proved some special cases. Here I give a combinatorial proof following Gian-Carlo Rota:
@phonner@johncarlosbaez@paulmasson The “symbolic method” (as in the lovely "Analytic Combinatorics" book by Flajolet and Sedgewick) gives a slick (after you buy into it, i.e. the background) proof of Dobiński's formula. I wrote a quick post about it here a while ago; don't know how understandable it is: https://shreevatsa.net/post/permutations-dobinski/
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