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.
Apéry number | binomial | binomial coefficient | central binomial coefficient | Christmas stocking theorem | Franel number | hypergeometric identity | hypergeometric series | idempotent number | Jonah formula | Klee's identity | Lucas correspondence theorem | married couples problem | Morley's formula | nexus number | Pascal's formula | Schmidt's problem | Stanley's identity | star of David theorem | Strehl identities | Székely identity | Waring formula | Worpitzky's identity
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