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%I #15 Mar 04 2014 13:47:52
%S 1,1,18,333,6318,122634,2429028,48974949,1002875094,20814628158,
%T 437088964860,9272342710962,198456435657036,4280758166952756,
%U 92972201833888200,2031520673763657621,44630859892110807654
%N Generalized Catalan numbers.
%H Fung Lam, <a href="/A068771/b068771.txt">Table of n, a(n) for n = 0..725</a>
%F a(n) = (9^n) * p(n, -8/9) with the row polynomials p(n, x) defined from array A068763.
%F a(n+1) = 9*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
%F G.f.: (1-sqrt(1-36*x*(1-8*x)))/(18*x).
%F Recurrence: (n+1)*a(n) = 288*(2-n)*a(n-2) + 18*(2*n-1)*a(n-1). - _Fung Lam_, Mar 04 2014
%F a(n) ~ sqrt(2) * 24^n / (3*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 04 2014
%t a[n_] := (288 (2 - n) a[n - 2] + 18 (2 n - 1) a[n - 1])/(n + 1); Table[a[n], {n, 0, 20}](* _Wesley Ivan Hurt_, Mar 04 2014 *)
%t CoefficientList[Series[(1-Sqrt[1-36*x*(1-8*x)])/(18*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 04 2014 *)
%Y Cf. A000108, A068764-A068770, A068772, A025227-A025230.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Mar 04 2002
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