The important binomial theorem states that sum_(k = 0)^n(n k) r^k = (1 + r)^n. Consider sums of powers of binomial coefficients a_n^(r) | congruent | sum_(k = 0)^n (n k)^r | = | _r F_(r - 1)(-n, ..., - n_︸_r ;1, ..., 1_︸_(r - 1) ;(-1)^(r + 1)), where _p F_q(a_1, ..., a_p ;b_1, ..., b_q ;z) 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.

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