5
$\begingroup$

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.

$\endgroup$
13
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.