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This should also be a comment. Most combinatorialists (for example, Enumerative Combinatorics, v.1, by R.P. Stanley, page 18) define the Stirling numbers of the first kind to be $(-1)^{(k-m)}c(k,m)$. s(k,m) := (-1)^{(k-m)}c(k,m)$. With that definition, you have the identity$\sum_{k \geq 0} S(n,k)s(k,m) = \delta_{n,m}$(ibid., p. 35). The sum you give does not always yield 0. When$n=2$and$m=1\$, for example, it equals 2.

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This should also be a comment. Most combinatorialists (for example, Enumerative Combinatorics, v.1, by R.P. Stanley, page 18) define the Stirling numbers of the first kind to be $(-1)^{(k-m)}c(k,m)$. With that definition, you have the identity $\sum_{k \geq 0} S(n,k)s(k,m) = \delta_{n,m}$ (ibid., p. 35). The sum you give does not always yield 0. When $n=2$ and $m=1$, for example, it equals 2.
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