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%I #15 Nov 20 2017 18:00:42
%S 0,1,7,59,544,5289,53256,549771,5782105,61698314,666014200,7257758425,
%T 79716116408,881431795012,9802065031740,109547572895811,
%U 1229634981583560,13855340183096319,156654090794892216
%N G.f. satisfies A(x) = x*(1+4A+A^2)^2/(1+A+A^2).
%H Alois P. Heinz, <a href="/A101487/b101487.txt">Table of n, a(n) for n = 0..400</a>
%H M. Bousquet-Mélou, <a href="http://arXiv.org/abs/math.CO/0501266">Limit laws for embedded trees</a>
%F a(n) ~ Gamma(1/4) * 2^(2*n - 2) * 3^(n + 1/2) / (Pi * n^(5/4)) * (1 - 2*sqrt(3*Pi) / (Gamma(1/4) * n^(1/4)) + 15*Pi*sqrt(2) / (4*Gamma(1/4)^2 * sqrt(n))). - _Vaclav Kotesovec_, Nov 20 2017
%p A:= proc(n) option remember; if n=0 then 0 else convert(series(x* (1+ 4*A(n-1) +A(n-1)^2)^2/ (1+A(n-1) +A(n-1)^2), x,n+1), polynom) fi end: a:= n-> coeff(A(n), x,n): seq(a(n), n=0..18); # _Alois P. Heinz_, Aug 23 2008
%t A[n_] := A[n] = If[n==0, 0, Normal[Series[x*(1+4*A[n-1]+A[n-1]^2)^2/(1+A[n-1]+ A[n-1]^2), {x, 0, n+1}]]]; a[n_] := Coefficient[A[n], x, n]; Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Mar 30 2015, after _Alois P. Heinz_ *)
%t CoefficientList[Series[(1 + Sqrt[1 - 12*x] - x*(8 + Sqrt[2]*Sqrt[(1 + Sqrt[1 - 12*x] - 2*(7 + 4*Sqrt[1 - 12*x])*x + 24*x^2) / x^2])) / (4*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Nov 20 2017 *)
%K nonn
%O 0,3
%A _Ralf Stephan_, Jan 21 2005
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