Question

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:

$$\pm\sum_{k}(-1)^k|s(n,2k)|=\mp|s(n,2)|\pm|s(n,4)|\mp|s(n,6)|\pm\ldots $$

or:

$$\pm\sum_{k}(-1)^k|s(n,2k+1)|=\pm|s(n,1)|\mp|s(n,3)|\pm|s(n,5)|\mp\ldots, $$

where $|s(n,k)|$ is an unsigned Stirling 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.

Answer 1

The generating function here is $$\sum_{n\geq 0}s(n,k) x^n y^k=\sum_{n\geq 0} \frac{x^n}{n!}y(y-1)\cdots (y-n+1)=e^{y\log(1+x)}.$$ If we put $y=i$ the coefficient of $x^n$ becomes $A_n+iB_n$ where $A_n$ and $B_n$ are your sequences. It is pretty clear from here that the exponential generating function for $A_n$ is $$\mathfrak {Re}\left(e^{i\log(1+x)}\right)=\cos(\log(1+x)),$$ and for $B_n$ it is $$\mathfrak{Im}\left(e^{i\log(1+x)}\right)=\sin(\log(1+x)).$$ Also these are A009454 and A003703.