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%I #13 Jan 25 2017 19:33:03
%S 1,2,14,116,1068,10512,108288,1152944,12586256,140119328,1584718432,
%T 18156868096,210302739712,2458400698368,28966931629056,
%U 343671253924608,4102106153148672,49225440239052288,593522724752742912,7186802279959262208,87357857306307234816,1065563386236346036224
%N G.f. satisfies: A(x) = (1 - 2*x*A(x))^2 * (3*A(x) - 2).
%H G. C. Greubel, <a href="/A231615/b231615.txt">Table of n, a(n) for n = 0..800</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = exp( x*(3*A(x)^2 - 2*A(x)) + Integral(3*A(x)^2 - 2*A(x) dx) ).
%F (2) A(x) = (1/x)*Series_Reversion( x*(1-6*x+6*x^2)/(1-2*x)^2 ).
%F (3) A(x) = 1 + 2*x*A(x)*(3*A(x) - 2)*(1 - x*A(x)).
%F (4) A(x) = 1 + Sum_{n>=2} 2^(n-2) * n * x^(n-1) * A(x)^n.
%F Recurrence: 3*n*(n+1)*(41*n-51)*a(n) = 3*n*(574*n^2 - 1001*n + 369)*a(n-1) - (1517*n^3 - 4921*n^2 + 4530*n - 1080)*a(n-2) + 8*(n-3)*(2*n-3)*(41*n-10)*a(n-3). - _Vaclav Kotesovec_, Dec 29 2013
%F a(n) ~ 2^(5*n-3/2) * 3^(2*n-5/2) / (n^(3/2) * sqrt(Pi*s) * r^n), where r = 22.00345492406548979... is the root of the equation -4478976 + 217728*r - 666*r^2 + r^3 = 0 and s = 0.00113473758620755613... is the root of the equation -41 + 30996*s + 4320054*s^2 + 181398528*s^3 = 0. - _Vaclav Kotesovec_, Dec 29 2013
%e G.f.: A(x) = 1 + 2*x + 14*x^2 + 116*x^3 + 1068*x^4 + 10512*x^5 +...
%e Related expansions.
%e (1 - 2*x*A(x))^2 = 1 - 4*x - 4*x^2 - 40*x^3 - 336*x^4 - 3120*x^5 -...
%e 3*A(x) - 2 = 1 + 6*x + 42*x^2 + 348*x^3 + 3204*x^4 + 31536*x^5 +...
%e 3*A(x)^2 - 2*A(x) = 1 + 8*x + 68*x^2 + 632*x^3 + 6252*x^4 + 64608*x^5 +...
%e log(A(x)) = 2*x + 24*x^2/2 + 272*x^3/3 + 3160*x^4/4 + 37512*x^5/5 +...
%t CoefficientList[1/x*InverseSeries[Series[x*(1-6*x+6*x^2)/(1-2*x)^2, {x, 0, 20}], x],x] (* _Vaclav Kotesovec_, Dec 29 2013 *)
%o (PARI) {a(n)=polcoeff((serreverse(x*(1-6*x+6*x^2)/(1-2*x)^2 +x^2*O(x^n))/x), n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1); for(i=1, n, A=exp(x*(3*A^2-2*A)+intformal(3*A^2-2*A +x*O(x^n)))); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1); for(i=1, n, A=1+2*x*A*(3*A-2)*(1-x*A) +x*O(x^n)); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A228966, A231552, A231553, A231554, A231556, A231616, A231618.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 11 2013
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