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 A086621 #13 Oct 23 2022 23:29:34
%S A086621 1,3,14,79,504,3514,26172,204831,1664696,13930840,119312544,
%T A086621 1041227642,9228614836,82867255956,752405060536,6897376441167,
%U A086621 63760133568096,593763928313128,5565678569009328,52475976165495960,497376657383374560,4736680863568248480,45304174896889357440
%N A086621 Main diagonal of square table A086620 of coefficients, T(n,k), of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xy*f(x,y)^2.
%H A086621 Andrew Howroyd, <a href="/A086621/b086621.txt">Table of n, a(n) for n = 0..500</a>
%F A086621 a(n) = Sum_{j=0..n} binomial(n,j) * binomial(2*n-j,n) * binomial(2*j,j)/(j+1). - _Andrew Howroyd_, Apr 11 2021
%F A086621 D-finite with recurrence n*(n-1)*(n+1)^2*a(n) -2*n*(n-1) *(4*n+3) *(2*n-1) *a(n-1) +4*(n-1) *(16*n^3-20*n^2-13*n+14) *a(n-2) -4*(n-2) *(4*n-9) *(4*n-3) *(n+1) *a(n-3)=0. - _R. J. Mathar_, Nov 02 2021
%F A086621 a(n) ~ sqrt(5) * 2^(2*n - 3/2) * phi^(2*n + 5/2) / (Pi * n^2), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Nov 19 2021
%p A086621 re:= sumtools:-sumrecursion(binomial(n,j) * binomial(2*n-j,n) * binomial(2*j,j)/(j+1),j,a(n)); # re = Mathar's recurrence
%p A086621 f:= gfun:-rectoproc({re = 0, a(0)=1, a(1)=3, a(2)=14}, a(n), remember): map(f, [$0..20]); # _Georg Fischer_, Oct 23 2022
%o A086621 (PARI) a(n) = {sum(j=0, n, binomial(n,j)*binomial(2*n-j, n)*binomial(2*j, j)/(j+1))} \\ _Andrew Howroyd_, Apr 11 2021
%Y A086621 Cf. A086620 (table), A086622 (antidiagonal sums).
%K A086621 nonn
%O A086621 0,2
%A A086621 _Paul D. Hanna_, Jul 24 2003
%E A086621 Terms a(18) and beyond from _Andrew Howroyd_, Apr 11 2021