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A finite summation

Let m,n be positive integers. Calculate

    \[\mathcal{S} = \sum_{k=1}^{2n} \prod_{i=0}^{m} \left ( \left \lfloor \frac{k+1}{2} \right \rfloor + \alpha + i \right )^{-1}\]

where \alpha is a non negative number and \left \lfloor x \right \rfloor represents the greatest integer less than or equal to x.

Solution

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