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 A068765 #19 Feb 03 2025 09:36:26
%S A068765 1,1,6,39,270,1962,14796,114831,911574,7368894,60457428,502162902,
%T A068765 4214515212,35686162548,304491863448,2615468845311,22598114065254,
%U A068765 196269877811574,1712578870493316,15005719955119698
%N A068765 Generalized Catalan numbers 3*x*A(x)^2 -A(x) +1 -2*x = 0.
%C A068765 a(n)=K(3,3; n)/3 with K(a,b; n) defined in a comment to A068763.
%H A068765 Fung Lam, <a href="/A068765/b068765.txt">Table of n, a(n) for n = 0..1000</a>
%F A068765 a(n)=(3^n)*p(n, -2/3) with the row polynomials p(n, x) defined from array A068763.
%F A068765 a(n+1)= 3*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
%F A068765 G.f.: (1-sqrt(1-12*x*(1-2*x)))/(6*x).
%F A068765 Recurrence: (n+1)*a(n) = 24*(2-n)*a(n-2) + 6*(2*n-1)*a(n-1). - _Fung Lam_, Mar 04 2014
%F A068765 a(n) ~ sqrt(6+6*sqrt(3)) * (6+2*sqrt(3))^n / (6*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 04 2014
%t A068765 CoefficientList[Series[(1-Sqrt[1-12*x*(1-2*x)])/(6*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 04 2014 *)
%Y A068765 Cf. A000108, A068764, A068766-72, A025227-30.
%K A068765 nonn,easy
%O A068765 0,3
%A A068765 _Wolfdieter Lang_, Mar 04 2002