Does someone know how to calculate sum $\sum_{k=1}^n \frac{1}{(n-k)!\cdot k} $? I was working something with matrices, i.e. calculating number of all cycles in Coates digraph, and got this weird expression(multiplied by $n!$) Thank you in advance.
2026-04-08 07:55:48.1775634948
Finite sum of reciprocal factorials
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I don't know of a closed form for your sequence, other than the hypergeometric
$$ {\frac {{\mbox{$_3$F$_1$}(1,1,1-n;\,2;\,-1)}}{ \left( n-1 \right) !}}$$
but the generating function is
$$ g(x) = \sum_{n=1}^\infty \sum_{k=1}^n \frac{x^n}{(n-k)! k} = -e^x \ln(1-x)$$