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 A357511 a(n) = numerator of Sum_{k = 1..n} (1/k) * binomial(n,k)^2 * binomial(n+k,k)^2 for n >= 1 with a(0) = 0 6
 0, 4, 54, 2182, 36625, 3591137, 25952409, 4220121443, 206216140401, 47128096330129, 1233722785504429, 364131107601152519, 9971452750252847789, 3611140187389794708497, 102077670374035974509597, 2922063451137950165057717, 169140610796591477659644439 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS A. Straub, Multivariate Apéry numbers and supercongruences of rational functions, arXiv:1401.0854 [math.NT] (2014). FORMULA Conjecture: a(p-1) == 0 (mod p^4) for all primes p >= 7 (checked up to p = 499). Note: the Apery numbers A(n) = A005259(n) = Sum_{k = 0..n} binomial(n,k)^2 * binomial(n+k,k)^2 satisfy the supercongruence A(p-1) == 1 (mod p^3) for all primes p >= 5 (see, for example, Straub, Introduction). EXAMPLE a(13 - 1) = 9971452750252847789 = (13^4)*37*2477*24197*157433 == 0 (mod 13^4). MAPLE seq(numer(add( (1/k) * binomial(n, k)^2 * binomial(n+k, k)^2, k = 1..n )), n = 0..20); PROG (PARI) a(n) = if (n, numerator(sum(k=1, n, binomial(n, k)^2*binomial(n+k, k)^2/k)), 0); \\ Michel Marcus, Oct 04 2022 CROSSREFS Cf. A005259, A357506, A357507, A357510, A357512, A357513. Sequence in context: A182264 A355128 A355126 * A094154 A168510 A324235 Adjacent sequences: A357508 A357509 A357510 * A357512 A357513 A357514 KEYWORD nonn,easy AUTHOR Peter Bala, Oct 01 2022 STATUS approved

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Last modified March 24 12:09 EDT 2023. Contains 361479 sequences. (Running on oeis4.)