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 A376466 #7 Sep 29 2024 09:20:23
%S A376466 1,3,127,9435,866751,89591753,9988439203,1173951006987,
%T A376466 143456999185855,18063466831218981,2329136571942011877,
%U A376466 306174745758226208537,40896708938016175140963,5536767359542664588001285,758259747093486125157272779,104880152366856305370319427435,14632959744552362547801104612799
%N A376466 a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)^2*A108625(n-1, k).
%C A376466 Cf. A005258(n) = Sum_{k = 0..n} (-1)^k*binomial(n, k)*binomial(n+k, k)^2.
%C A376466 The sequence of Apéry numbers A005258 satisfies the pair of supercongruences
%C A376466 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 A376466 and
%C A376466 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 A376466 We conjecture that the present sequence satisfies the same pair of supercongruences. Some examples are given below.
%F A376466 Examples of supercongruences:
%F A376466 a(11) - a(1) = 306174745758226208537 - 3 = 2*(11^3)*17*79367*85245689663 == 0 (mod 11^3).
%F A376466 a(10) - a(0) = 2329136571942011877 - 1 = (2^2)*(11^3)*17011*25717400209 == 0 (mod 11^3).
%p A376466 A108625(n, k) := add( binomial(n, i)^2 * binomial(n+k-i, k-i), i = 0..k):
%p A376466 a(n) := add((-1)^(n+k)*binomial(n, k)*binomial(n+k, k)^2*A108625(n-1, k), k = 0..n):
%p A376466 seq(a(n), n = 0..25);
%Y A376466 Cf. A005258, A376458 - A376465.
%K A376466 nonn,easy
%O A376466 0,2
%A A376466 _Peter Bala_, Sep 25 2024