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 A357561 #13 Oct 08 2022 10:19:27
%S A357561 0,4,-27,1367,-15625,3129353,-14749,308477847,14343020119,
%T A357561 80826490175689,618729030402659,6526775794564145231,
%U A357561 52975460244520902439,965428117884339747694757,8161435689582967449592663,70159702295938799645630801,4897311439674525483507166097,212741477113936719632186271679919
%N A357561 a(n) = the numerator of ( Sum_{k = 1..n} (-1)^(n+k)*(1/k^3)*binomial(n,k)* binomial(n+k,k)^2 ).
%C A357561 Define S_m(n) = the numerator of Sum_{k = 1..n} (-1)^(n+k)*(1/k^m)*binomial(n,k)* binomial(n+k,k)^2, so that S_0(n) = -1 + A005258(n), one type of Apéry numbers. The present sequence is the case m = 3. See A357560 for the case m = 1.
%C A357561 Conjecture: for odd m >= 1, S_m(p-1) == 0 (mod p^4) for all primes p > m+4.
%H A357561 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 A357561 Conjecture: a(p-1) == 0 (mod p^4) for p = 5 and all primes p >= 11 (checked up to p = 499).
%F A357561 Note: the Apéry numbers B(n) = A005258(n) = Sum_{k = 0..n} (-1)^(n+k)* binomial(n,k)*binomial(n+k,k)^2 satisfy the supercongruences B(p-1) == 1 (mod p^3) for all primes p >= 5 (see, for example, Straub, Example 3.4).
%e A357561 Example of a supercongruence:
%e A357561 p = 17: a(17 - 1) = 212741477113936719632186271679919 = (17^4)*4871421029* 12036670481533 == 0 (mod 17^4).
%p A357561 seq( numer(add( (-1)^(n+k) * (1/k^3) * binomial(n,k) * binomial(n+k,k)^2, k = 1..n )), n = 0..20 );
%Y A357561 Cf. A005258, A357510, A357511, A357512, A357513, A357558, A357559, A357560.
%K A357561 sign,easy
%O A357561 0,2
%A A357561 _Peter Bala_, Oct 04 2022