I am trying to evaluate a sum over binomial coefficients which is giving me some problems. Specifically I want to calculate:


My main thought was to convert the fraction here into:

$$\int_0^1 x^{c-r}dx,$$ move the integral out of the sum, alongside the $x^c$ and then attempt to rewrite as some closed form function. I however, cannot see what the generating function should be.

Note, my aim here is to avoid having a sum - some product of binomial coefficients would be ideal but I obviously do not know if this exists!

Any help on summing this would be greatly appreciated.


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