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I'm aware of the identity \begin{align} \sum_{i = 0}^{k} i! \binom{n+1}{i + 1} S(k,i) = H_{n,-k}, \end{align} where $H_{n,-k}$ is a generalized Harmonic number defined by $H_{n,m} = \sum_{r = 1}^{n} r^{-m}$. I believe the following sum is related to the generalized Harmonic numbers as well, \begin{align} \sum_{i = 0}^{k} (-1)^{i} i! \binom{n-i}{k-i} S(j,i) \end{align} and should be a nice function of $n, k$ and $j$, where $0 \leq j, k \leq n$. Any hints are certainly welcome!

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