Does anybody know of any identities or combinatorial interpretations for alternating sums of alternate Stirling numbers?
I am particularly interested in expressions of the form:
$$\pm\sum_{k\text{ even}}(-1)^{\frac{k}{2}}|s(n,k)|=\mp|s(n,2)|\pm|s(n,4)|\mp|s(n,6)|\pm\ldots $\pm\sum_{k}(-1)^k|s(n,2k)|=\mp|s(n,2)|\pm|s(n,4)|\mp|s(n,6)|\pm\ldots $$
or:
$$\pm\sum_{k\text{ odd}}(-1)^{\frac{k-1}{2}}|s(n,k)|=\pm|s(n,1)|\mp|s(n,3)|\pm|s(n,5)|\mp\ldots, $\pm\sum_{k}(-1)^k|s(n,2k+1)|=\pm|s(n,1)|\mp|s(n,3)|\pm|s(n,5)|\mp\ldots, $$
where the $|s(n,k)|$ are is an unsigned Stirling numbers number of the first kind.
However, I would be happy with any related identities or information (including an argument as to why sums like this might not have nice formulae or count anything interesting). As far as I can tell these do not appear on the OEIS, or in any of the literature.
