The Franel numbers are the numbers Fr_n = sum_(k = 0)^n (n k)^3, where (n k) is a binomial coefficient. The first few values for n = 0, 1, ... are 1, 2, 10, 56, 346, ... (OEIS A000172). They arise in the first Strehl identity sum_(k = 0)^n (n k)^3 = sum_(k = 0)^n (n k)^2(2k n), and can be written in closed form as Fr_n = _3 F_2(-n, - n, - n;1, 1;-1), where _3 F_2(a, b, c;d, e;z) is a generalized hypergeometric function.
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