A068768 - OEIS

%I #21 Feb 03 2025 09:36:38

%S 1,1,12,150,1944,25992,356832,5008824,71629920,1040509152,15315578496,

%T 227981324736,3426473187072,51929043390720,792725911280640,

%U 12178706839758720,188158789025809920,2921622674591946240

%N Generalized Catalan numbers 6*x*A(x)^2 -A(x) +1 -5*x =0.

%C a(n) = K(6,6; n)/6 with K(a,b; n) defined in a comment to A068763.

%H Fung Lam, <a href="/A068768/b068768.txt">Table of n, a(n) for n = 0..810</a>

%F a(n) = (6^n) * p(n, -5/6) with the row polynomials p(n, x) defined from array A068763.

%F a(n+1) = 6*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).

%F G.f.: (1-sqrt(1-24*x*(1-5*x)))/(12*x).

%F D-finite with recurrence: (n+1)*a(n) = 120*(2-n)*a(n-2) + 12*(2*n-1)*a(n-1). - _Fung Lam_, Mar 04 2014

%F a(n) ~ sqrt(3+3*sqrt(6)) * (12+2*sqrt(6))^n / (6*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 04 2014

%t CoefficientList[Series[(1-Sqrt[1-24*x*(1-5*x)])/(12*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 04 2014 *)

%Y Cf. A000108, A068764-A068767, A068769-A068772, A025227-A025230.

%K nonn,easy

%O 0,3

%A _Wolfdieter Lang_, Mar 04 2002