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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 ...

I stumbled upon the following inequality which, I believe, is true. I was able to prove it for small k, but I have no proof for the general case. Any help is welcome. Let $n\geq k\geq 1$ then $$\left(...

Let $S(n, j)$ be Stirling's number of second kind. Let $p\in [0,1]$ and $m \in N$. Bound from above the following sum: $$ \sum_{j=0}^m S(n,j) {m \choose j}\, j! \, p^j $$