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.



Alternating sums of alternate Stirling numbers 
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](http://oeis.org/A009454) and [A003703](http://oeis.org/A003703).


