A calculation about the sum of the product of Binomial and Stirling numbers of the first kind

Question

I have came to a calculation about the sum of the product of Binomial and Stirling first numbers as following $$ \sum_{i=0}^{2k-j}\binom{2k}{i}(-1)^{i}s(2k-i,2k-i-j)~\text{for}~j=0,1,2,\ldots,k-1, ~\text{and}~k\in \mathbb{N}^{+}. $$ where $s(n,k)$ is the Stirling number of the first kind with $x(x+1)\cdots(x+n-1)=\sum_{k=0}^{n}s(n,k)x^k$. I just tested some special cases, finding that the summation leads to 0, how to prove it or relate to the expansion of some function series? Can anyone give some insights about this equation?

Answer 1

Your identity has been studied before for example here On Certain Sums of Stirling Numbers with Binomial Coefficients by Gould et. al

In particular, your numbers are $f_{2(k-j)}(j)$ which by Theorem 4 in the paper are evaluated to $0.$

You can also check that they count certain derangements in the general case.