Find a closed form formula for this sum: $$\sum_{1\le i<j\le m} \sum_{\substack{1\le k,l\le n \\ k+l\le n}}{n\choose k} {n-k\choose l} (j-i-1)^{n-k-l}$$
It's quite likely that it can be done with standard techniques, but I haven't tried to do it completely, because it was very tedious. I think combinatorial interpretation can be a nice way here, although tricky. It's easy to prove that inner sum is equal to $n(n-1)(j-i+1)^{n-2}$ but then I really doubt sum $\sum_{1\le i<j\le m}n(n-1)(j-i+1)^{n-2}$ can be easily computed. Can anybody help?