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%I #26 May 24 2022 11:51:57
%S 1,1,-6,33,-162,666,-1836,-2079,79542,-741474,4907628,-24837030,
%T 82449900,53319060,-3741922008,38613958497,-274566158298,
%U 1475669401398,-5211777090564,-2356585871778,240686500011588,-2593621485808596,19047621883804056,-105353643788834598
%N Expansion of g.f.: (-1 + sqrt(1+12*x+48*x^2)) / (6*x).
%C This sequence is the member (q=-3) of a class of generalized Catalan numbers (see A000108), with g.f. (1-sqrt(1-q*4*x*(1-(q-1)*x)))/(2*q*x), q<>0.
%H Fung Lam, <a href="/A240880/b240880.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: (-1 + sqrt(1+12*x+48*x^2)) / (6*x).
%F D-finite with recurrence: (n+3)*a(n+2)+6*(2*n+3)*a(n+1)+48*n*a(n)=0, a(0)=1, a(1)=1.
%F Lim sup n->infinity |a(n)|^(1/n) = 4*sqrt(3) = 6.9282... - _Vaclav Kotesovec_, May 02 2014
%F a(n) ~ 3^(n/2-1)*4^n / (n^(3/2)*sqrt(Pi)) * (sqrt(3)*cos(5*Pi*n/6) + 3*sin(5*Pi*n/6) - (15*sqrt(3)*cos(5*Pi*n/6) + 9*sin(5*Pi*n/6))/(8*n)). - _Vaclav Kotesovec_, May 02 2014
%t CoefficientList[Series[(Sqrt[1+12x+48x^2]-1)/(6x),{x,0,30}],x] (* _Harvey P. Dale_, May 24 2022 *)
%Y Cf. A000108, A258723.
%K sign,easy
%O 0,3
%A _Fung Lam_, May 01 2014
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