I want to compute/prove that the following sum is positive: $$ \sum_{i = 0}^n \left[\frac{D(n - i, i)}{d} \sum_{j = m}^d s(d, j) \binom{j}{m} (d - i)^{j - m}\right] > 0 $$ where $s(d, j)$ is the stirling number of the first kind, $D(n - i, i)$ is the Delannoy number, $d = 2 n + 2\geq 4$ is an even number, and $0 \leq m \leq d$. The first step could be computing either $$ a_i = \sum_{j = m}^d s(d, j) \binom{j}{m} (d - i)^{j - m} $$ which seems to be a big positive number when $i = 0$ and then alternates signs as i increases. I suspect that $a_i$ could be obtained by exponential generating functions, or that it could be the coefficient of a Fourier transform as in this question, but I haven't found it.
Or compute: $$ d_j = \sum_{i = 0}^n \frac{D(n - i, i)}{d} (d - i)^{j - m} $$ Intuitively, a closed formula is unlikely so some sophisticated estimation would do the job, but I'm quite inexperienced in that. Thanks!
