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Binomial sums

The important binomial theorem states that(1)Consider sums of powers of binomial coefficients(2)(3)where is a generalized hypergeometric function. When they exist, the recurrence equations that give solutions to these equations can be generated quickly using Zeilberger's algorithm.For , the closed-form solution is given by(4)i.e., the powers of two. obeys the recurrence relation(5)For , the closed-form solution is given by(6)i.e., the central binomial coefficients. obeys the recurrence relation(7)Franel (1894, 1895) was the first to obtain recurrences for ,(8)(Riordan 1980, p. 193; Barrucand 1975; Cusick 1989; Jin and Dickinson 2000), so are sometimes called Franel numbers. The sequence for cannot be expressed as a fixed number of hypergeometric terms (Petkovšek et al. 1996, p. 160), and therefore has no closed-form hypergeometric expression.Franel (1894, 1895) was also the first to obtain the recurrence..

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