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%I #16 Apr 17 2014 00:41:20
%S 1,1,6,39,270,1962,14796,114831,911574,7368894,60457428,502162902,
%T 4214515212,35686162548,304491863448,2615468845311,22598114065254,
%U 196269877811574,1712578870493316,15005719955119698
%N Generalized Catalan numbers.
%C a(n)=K(3,3; n)/3 with K(a,b; n) defined in a comment to A068763.
%H Fung Lam, <a href="/A068765/b068765.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n)=(3^n)*p(n, -2/3) with the row polynomials p(n, x) defined from array A068763.
%F a(n+1)= 3*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
%F G.f.: (1-sqrt(1-12*x*(1-2*x)))/(6*x).
%F Recurrence: (n+1)*a(n) = 24*(2-n)*a(n-2) + 6*(2*n-1)*a(n-1). - _Fung Lam_, Mar 04 2014
%F a(n) ~ sqrt(6+6*sqrt(3)) * (6+2*sqrt(3))^n / (6*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 04 2014
%t CoefficientList[Series[(1-Sqrt[1-12*x*(1-2*x)])/(6*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 04 2014 *)
%Y Cf. A000108, A068764, A068766-72, A025227-30.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Mar 04 2002
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