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%I #15 Oct 08 2022 10:10:04
%S 0,4,54,2182,36625,3591137,25952409,4220121443,206216140401,
%T 47128096330129,1233722785504429,364131107601152519,
%U 9971452750252847789,3611140187389794708497,102077670374035974509597,2922063451137950165057717,169140610796591477659644439
%N 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
%H A. Straub, <a href="https://arxiv.org/abs/1401.0854">Multivariate Apéry numbers and supercongruences of rational functions</a>, arXiv:1401.0854 [math.NT] (2014).
%F Conjecture: a(p-1) == 0 (mod p^4) for all primes p >= 7 (checked up to p = 499).
%F 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).
%e a(13 - 1) = 9971452750252847789 = (13^4)*37*2477*24197*157433 == 0 (mod 13^4).
%p seq(numer(add( (1/k) * binomial(n,k)^2 * binomial(n+k,k)^2, k = 1..n )), n = 0..20);
%o (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
%Y Cf. A005259, A357506, A357507, A357510, A357512, A357513.
%K nonn,easy
%O 0,2
%A _Peter Bala_, Oct 01 2022
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