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%I #9 Nov 27 2017 18:24:52
%S 1,3,6,36,249,1932,16044,139500,1253934,11558316,108658902,1037800920,
%T 10041891132,98230257636,969814634424,9651213968784,96710160474513,
%U 974967422602428,9881687141571732,100632995795535588
%N G.f. satisfies: 13*A(x) = 12 + 27*x + A(x)^4, starting with [1,3,6].
%C See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.
%F G.f.: A(x) = 1 + Series_Reversion((1+13*x - (1+x)^4)/27).
%F G.f.: A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (12 + 27*x)^(3*n+1) / 13^(4*n+1). - _Paul D. Hanna_, Jan 24 2008
%F a(n) ~ 2^(3*n - 7/3) * 3^(2*n) / (13^(1/3) * sqrt(Pi) * n^(3/2) * (2^(1/3)*13^(4/3) - 32)^(n - 1/2)). - _Vaclav Kotesovec_, Nov 27 2017
%e A(x) = 1 + 3*x + 6*x^2 + 36*x^3 + 249*x^4 + 1932*x^5 +...
%e A(x)^4 = 1 + 12*x + 78*x^2 + 468*x^3 + 3237*x^4 + 25116*x^5 +...
%t CoefficientList[1 + InverseSeries[Series[(1+13*x - (1+x)^4)/27, {x, 0, 20}], x], x] (* _Vaclav Kotesovec_, Nov 27 2017 *)
%o (PARI) {a(n)=local(A=1+3*x+6*x^2+x*O(x^n));for(i=0,n,A=A+(-13*A+12+27*x+A^4)/9);polcoeff(A,n)}
%Y Cf. A120588 - A120594, A120596 - A120607; A245043.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 16 2006
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