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 A376463 #8 Sep 29 2024 09:20:10
%S A376463 1,13,937,110173,16431001,2815533013,528281347609,105661979187421,
%T A376463 22160058768609049,4820836639111911013,1079739020625352737937,
%U A376463 247635383880853678809541,57923551410778898112945769,13775523966484086307239139141,3322958149149086403877851762937,811467759428066412526078761086173
%N A376463 a(n) = Sum_{k = 0..n} binomial(n, k)^2*binomial(n+k, k)^2*A108625(n, k).
%C A376463 The sequence of Apéry numbers A005259 defined by A005259(n) = Sum_{k = 0..n} binomial(n, k)^2*binomial(n+k, k)^2 satisfies the pair of supercongruences
%C A376463 1) A005259(n*p^r) == A005259(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r
%C A376463 and
%C A376463 2) A005259(n*p^r - 1) == A005259(n*p^(r-1) - 1) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r.
%C A376463 We conjecture that the present sequence satisfies the same pair of supercongruences. Some examples are given below.
%e A376463 Examples of supercongruences:
%e A376463 a(11) - a(1) = 247635383880853678809541 - 13 = (2^3)*(3^2)*(11^3)*13*178697* 389023*2859343 == 0 (mod 11^3).
%e A376463 a(10) - a(0) = 1079739020625352737937 - 1 = (2^4)*3*(11^3)*13*1459*601831*1480561 == 0 (mod 11^3).
%p A376463 A108625(n, k) := add(binomial(n, i)^2 * binomial(n+k-i, k-i), i = 0..k):
%p A376463 a(n) := add(binomial(n, k)^2*binomial(n+k, k)^2*A108625(n, k), k = 0..n):
%p A376463 seq(a(n), n = 0..25);
%Y A376463 Cf. A005259, A376458 - A376466.
%K A376463 nonn,easy
%O A376463 0,2
%A A376463 _Peter Bala_, Sep 24 2024