I have the following set of series involving the Stirling numbers 1'st kind and binomials, which can be understood as a set of dot-products of row- and column-vectors of two infinite matrices (where R and C indicate rows and columns, beginning at zero):
$$ w_{R,C} =\sum_{k=\max(R,C)}^\infty (-1)^k {s_1(1+k,1+k-R)\over k!} \cdot (-1)^C (1+C)^k \cdot \binom {1+k}{1+C} $$
I've tested this heuristically for several R and C and always approximated zero; also wolfram-alpha can evaluate this explicitely to zero if feeded with
sum (-1)^k * StirlingS1(k+1,1+k-R)/k! * (1+C)^k * binomial(1+k,1+C), for k=max(C,R) to infty
where we replace $C$, $R$ and $\max(C,R)$ with actual values.
However, I've no option to let wolfram-alpha answer this in general.
I've proved this for $C=0$ and the first few R using exponential generating functions, but again, a general proof is out of reach for me (possibly I'm overlooking something trivial like telescoping...), so I ask for help here.
The convention for Stirling numbers first kind as in Math'ica, indexes beginning at zero:
$ \small \qquad \qquad \begin{array} {rrrrr} 1 & . & . & . & . & . \\ 0 & 1 & . & . & . & . \\ 0 & -1 & 1 & . & . & . \\ 0 & 2 & -3 & 1 & . & . \\ 0 & -6 & 11 & -6 & 1 & . \\ 0 & 24 & -50 & 35 & -10 & 1 \end{array} $
If some background is of interest: here are the questions on MSE
http://math.stackexchange.com/questions/16228 // question of some user which motivated me to look at an example
http://math.stackexchange.com/questions/89853 // my follow-up question dealing with the current problem
and a more worked out treatize on this in a pdf-file http://go.helms-net.de/math/divers/InverseNullmatrix.pdf
editorial remark: I had a typo in the first version of this question in the first formula and in the wolfram-alpha-formula which I corrected here and also in the .pdf-file. That was induced by the seemingly recent changes of evaluation-method at wolfram-alpha, which do no more evaluate the expression to the exact, but only to an approximate value (in the cost-free version); this is different from the behave in december 2011 when I first wrote the .pdf-file.
