This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A357511 #17 Jun 02 2025 15:26:15
%S A357511 0,4,54,2182,36625,3591137,25952409,4220121443,206216140401,
%T A357511 47128096330129,1233722785504429,364131107601152519,
%U A357511 9971452750252847789,3611140187389794708497,102077670374035974509597,2922063451137950165057717,169140610796591477659644439
%N 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.
%H A357511 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 A357511 Conjecture: a(p-1) == 0 (mod p^4) for all primes p >= 7 (checked up to p = 499).
%F A357511 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 A357511 a(13 - 1) = 9971452750252847789 = (13^4)*37*2477*24197*157433 == 0 (mod 13^4).
%p A357511 seq(numer(add( (1/k) * binomial(n,k)^2 * binomial(n+k,k)^2, k = 1..n )), n = 0..20);
%o A357511 (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 A357511 Cf. A005259, A357506, A357507, A357510, A357512, A357513.
%K A357511 nonn,easy
%O A357511 0,2
%A A357511 _Peter Bala_, Oct 01 2022