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Is there a closed form for the sum below?

$$\sum_{s=0}^{m-1} \sum_{t=0}^{m-1} s~t~(m-s)~(m-t)\left|s-t\right|$$

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2 Answers 2

Yes, there is. Split the sum as follows: $$\sum_{s=0}^{m-1} \sum_{t=0}^{m-1} st(m-s)(m-t)\left\vert s-t\right \vert = \sum_{s=0}^{m-1} \left( \underbrace{\sum_{t=0}^{s} st(m-s)(m-t)(s-t)}_{\dfrac{s^2(s^2-1)(m-s)(2m-s)}{12}} + \underbrace{\sum_{t=s+1}^{m-1} st(m-s)(m-t)(t-s)}_{\dfrac{s((m-s)^2-1)(m-s)^2(m+s)}{12}} \right)$$ Now the summation over $s$ can again be evaluated since we need only sums of the form $\displaystyle \sum_{s=0}^{m-1} s^k$ where $k \in \{1,2,3,4,5,6\}$.

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Using maple I obtained the value of the sum as $$\frac{(m+2)(m+1)m(m-1)(m-2)(3m^2+1)}{420}.$$ This is more symmetric than I would have thought.

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