Apéry's numbers are defined by A_n | = | sum_(k = 0)^n (n k)^2 (n + k k)^2 | = | sum_(k = 0)^n [(n + k)!]^2/((k!)^4 [(n - k)!]^2) | = | _4 F_3(-n, - n, n + 1, n + 1;1, 1, 1;1), where (n k) is a binomial coefficient. The first few for n = 0, 1, 2, ... are 1, 5, 73, 1445, 33001, 819005, ... (OEIS A005259).
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