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 A376465 #7 Sep 29 2024 09:20:18
%S A376465 1,5,289,32909,4846001,824723005,154077084505,30725590395197,
%T A376465 6429986799134257,1396511202452212733,312375607924873231289,
%U A376465 71567605248444374973205,16725718218774077760354953,3974882968204513021199112653,958241214962583413382405975337,233879553339714596653197104362909
%N A376465 a(n) = Sum_{k = 0..n} binomial(n, k)^2*binomial(n+k, k)^2*A108625(n-1 k).
%C A376465 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 A376465 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 A376465 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 A376465 We conjecture that the present sequence satisfies the same pair of supercongruences. Some examples are given below.
%e A376465 Examples of supercongruences:
%e A376465 a(11) - a(1) = 71567605248444374973205 - 5 = (2^4)*(5^2)*(11^3)*19*56040893*126246629 == 0 (mod 11^3).
%e A376465 a(10) - a(0) = 312375607924873231289 - 1 = (2^3)*(11^3)*17*1725679541724893 == 0 (mod 11^3).
%p A376465 A108625(n, k) := add(binomial(n, i)^2 * binomial(n+k-i, k-i), i = 0..k):
%p A376465 a(n) := add(binomial(n, k)^2*binomial(n+k, k)^2*A108625(n-1, k), k = 0..n):
%p A376465 seq(a(n), n = 0..25);
%Y A376465 Cf. A005259, A376458 - A376466.
%K A376465 nonn,easy
%O A376465 0,2
%A A376465 _Peter Bala_, Sep 25 2024