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%I #14 Jun 28 2014 01:06:14
%S 1,2,10,66,570,6162,80010,1212066,20983770,408687282,8844164010,
%T 210530630466,5467167038970,153805368043602,4659779072312010,
%U 151259403573751266,5237308594356166170,192673897986624475122,7505181282611209004010,308589102795660836942466
%N E.g.f.: (2 + exp(3*x)) / (4 - exp(3*x)).
%F E.g.f. A(x) satisfies:
%F (1) A'(x) = -1 + A(x) + 2*A(x)^2.
%F (2) A(x) = exp(x + Integral 2*A(x) - 1/A(x) dx).
%F (3) A(x) = 1 + Series_Reversion( Integral 1/((2+x)*(1+2*x)) dx ).
%F a(n) ~ n!/2 * (3/(2*log(2)))^(n+1). - _Vaclav Kotesovec_, Dec 19 2013
%e E.g.f.: A(x) = 1 + 2*x + 10*x^2/2! + 66*x^3/3! + 570*x^4/4! + 6162*x^5/5! +...
%t CoefficientList[Series[(2+E^(3*x))/(4-E^(3*x)), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Dec 19 2013 *)
%o (PARI) {a(n)=local(A=1+x,X=x+x*O(x^n));n!*polcoeff((2+exp(3*X))/(4-exp(3*X)),n)}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(x+intformal(2*A-1/A+x*O(x^n)))); n!*polcoeff(A, n)}
%o for(n=0,20,print1(a(n), ", "))
%Y Cf. A190818.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 18 2013
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