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 A183068 #22 Jul 16 2024 15:41:04
%S A183068 1,26,3246,606500,137915470,35218238076,9702014515116,
%T A183068 2818627826459016,851612982884556750,265166341958122567820,
%U A183068 84556145346599067308596,27489903606068331188121816,9081510922185418532993154796
%N A183068 Central terms of triangle A183065.
%H A183068 Peter Bala, <a href="/A183068/a183068.pdf">A recurrence for A183068</a>
%F A183068 a(n) = A183065(2*n,n).
%F A183068 a(n) = [x^(2*n)*y^n] Sum_{m >= 0} (4*m)!/m!^4 * x^(2*m)*y^m/(1-x-x*y)^(4*m+1).
%F A183068 From _Peter Bala_, Jul 15 2024: (Start)
%F A183068 a(n) = binomial(2*n, n)*Sum_{k = 0..n} binomial(n, k)^2*binomial(2*n+2*k, 2*k)* binomial(2*k, k) = Sum_{k = 0..n} (2*n+2*k)!/(k!^4*(n-k)!^2). Cf. A002897(n) = Sum_{k = 0..n} (2*n+k)!/(k!^3*(n-k)!^2) and A005258(n) = n!*Sum_{k = 0..n} (n+k)!/(k!^3*(n-k)!^2).
%F A183068 a(n) = binomial(2*n, n)*hypergeom([-n, -n, n+1, n+1/2], [1, 1, 1], 4).
%F A183068 Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for all primes p and positive integers n and r (End)
%F A183068 a(n) ~ sqrt(3) * (2 + sqrt(6))^(4*n + 3/2) / (16*Pi^2*n^2). - _Vaclav Kotesovec_, Jul 16 2024
%p A183068 seq(simplify(binomial(2*n, n)*hypergeom([-n, -n, n+1, n+1/2], [1, 1, 1], 4)), n = 0..20);
%o A183068 (PARI) {a(n)=polcoeff(polcoeff(sum(m=0,2*n,(4*m)!/m!^4*x^(2*m)*y^m/(1-x-x*y+x*O(x^(2*n)))^(4*m+1)),2*n,x),n,y)}
%Y A183068 Cf. A183065, A183066, A183067.
%K A183068 nonn,easy
%O A183068 0,2
%A A183068 _Paul D. Hanna_, Dec 22 2010