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A068772 Generalized Catalan numbers 10*x*A(x)^2 -A(x) +1 -9*x =0.

%I A068772 #17 Feb 03 2025 09:36:07
%S A068772 1,1,20,410,8600,184200,4020000,89205000,2008700000,45816140000,
%T A068772 1056825200000,24618524200000,578457724000000,13695679012000000,
%U A068772 326448619920000000,7827776361090000000,188701194087000000000
%N A068772 Generalized Catalan numbers 10*x*A(x)^2 -A(x) +1 -9*x =0.
%C A068772 This is the tenth member in the a-family of sequences K(a,a; n), a=1,2,3,...,n>=0, defined in a comment to the array A068763.
%H A068772 Fung Lam, <a href="/A068772/b068772.txt">Table of n, a(n) for n = 0..700</a>
%F A068772 a(n) = (10^n) * p(n, -9/10) with the row polynomials p(n, x) defined from array A068763.
%F A068772 a(n+1) = 10*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
%F A068772 G.f.: (1-sqrt(1-40*x*(1-9*x)))/(20*x).
%F A068772 Recurrence: (n+1)*a(n) = 360*(2-n)*a(n-2) + 20*(2*n-1)*a(n-1). - _Fung Lam_, Mar 05 2014
%F A068772 a(n) ~ sqrt(5+5*sqrt(10)) * (20+2*sqrt(10))^n / (10*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 06 2014
%t A068772 a[0] = 1; a[1] = 1; a[n_] := (360 (2 - n) a[n - 2] + 20 (2 n - 1) a[n - 1])/(n + 1); Table[a[n], {n, 0, 20}] (* _Wesley Ivan Hurt_, Mar 04 2014 *)
%t A068772 CoefficientList[Series[(1-Sqrt[1-40*x*(1-9*x)])/(20*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 06 2014 *)
%Y A068772 Cf. A000108, A068764-A068771, A025227-A025230.
%K A068772 nonn,easy
%O A068772 0,3
%A A068772 _Wolfdieter Lang_, Mar 04 2002