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 A376462 #9 Sep 29 2024 09:19:22
%S A376462 1,5,109,3317,121501,4954505,216867925,9981053045,476860000285,
%T A376462 23451310381505,1180189308268609,60519806861966105,
%U A376462 3152285573768063461,166371462775232899553,8880340127444426907109,478649327347386225075317,26019989011889817463755805,1425143757811438999747555313,78578956793385528989609594089
%N A376462 a(n) = Sum_{k = 0..n} binomial(n, k)^2*binomial(n+k, k)*A108625(n, n-k).
%C A376462 The sequence of Apéry numbers A005258 defined by A005258(n) = Sum_{k = 0..n} binomial(n, k)^2*binomial(n+k, k) satisfies the pair of supercongruences
%C A376462 1) A005258(n*p^r) == A005258(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r
%C A376462 and
%C A376462 2) A005258(n*p^r - 1) == A005258(n*p^(r-1) - 1) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r.
%C A376462 We conjecture that the present sequence satisfies the same pair of supercongruences. Some examples are given below.
%e A376462 Examples of supercongruences:
%e A376462 a(11) - a(1) = 60519806861966105 - 5 = (2^2)*(3^2)*(5^2)*(11^3)*197*256454747 == 0 (mod 11^3).
%e A376462 a(10) - a(0) = 1180189308268609 - 1 = (2^6)*3*(11^3)*37*2789*44753 == 0 (mod 11^3).
%p A376462 A108625(n, k) := add(binomial(n, i)^2 * binomial(n+k-i, k-i), i = 0..k):
%p A376462 a(n) := add(binomial(n, k)^2*binomial(n+k, k)*A108625(n, n-k), k = 0..n):
%p A376462 seq(a(n), n = 0..25);
%Y A376462 Cf. A005258, A376458 - A376466.
%K A376462 nonn,easy
%O A376462 0,2
%A A376462 _Peter Bala_, Sep 24 2024