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 A375177 #12 Aug 14 2024 13:12:51
%S A375177 1,5,26,134,670,3262,15540,72732,335478,1528670,6894316,30820660,
%T A375177 136736236,602610764,2640266600,11508115320,49928451750,215717144670,
%U A375177 928515985980,3983029119300,17032882625220,72631992447300,308911087394520,1310670689270280,5548646191175100
%N A375177 a(n) = 1/(n + 1)^2 * Sum_{k = 1..n+1} (k^4)*binomial(n+1, k)^2.
%C A375177 Compare with the identity 1/(n+1)^2 * Sum_{k = 1..n+1} (k^2)*binomial(n+1, k)^2 = binomial(2*n, n) = A000984(n).
%C A375177 The central binomial coefficients satisfy the supercongruence binomial(2*p, p) == 2 (mod p^3) for all primes p >= 5 (Wolstenholme's theorem).
%H A375177 Romeo Meštrović, <a href="https://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012)</a>, arXiv:1111.3057 [math.NT], (2011).
%F A375177 a(n) = hypergeom([2, 2, -n, -n], [1, 1, 1], 1).
%F A375177 a(n) = (1/2) * P(n)/(2*n - 1) * binomial(2*n, n), where P(n) = n^3 + 4*n^2 + 2*n - 2.
%F A375177 P-recursive: n*P(n-1)*a(n) = 2*(2*n - 3)*P(n)*a(n-1) with a(1) = 1.
%F A375177 a(p) == 2*p + 2 (mod p^3) for all primes p >= 5.
%p A375177 seq( 1/(n+1)^2 *add( (k^4)*binomial(n+1, k)^2, k = 1..n+1), n = 0..25);
%p A375177 # faster program for large n
%p A375177 a := proc (n) option remember; if n = 0 then 1 else (4*n-6)*(n^3+4*n^2+2*n-2)*a(n-1)/(n*(n^3+n^2-3*n-1)) end if; end:
%p A375177 seq(a(n), n = 0..25);
%Y A375177 Cf. A000984, A092443.
%K A375177 nonn,easy
%O A375177 1,2
%A A375177 _Peter Bala_, Aug 02 2024