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%I #7 Nov 28 2017 03:53:58
%S 1,3,15,210,3510,65562,1310901,27446760,594104940,13187589690,
%T 298555767279,6867021319722,160017552201780,3769622456958720,
%U 89628027015591870,2148034269252052608,51836638064282565579
%N G.f. satisfies: 15*A(x) = 14 + 27*x + A(x)^6, starting with [1,3,15].
%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+15*x - (1+x)^6)/27). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(6*n,n)/(5*n+1) * (14+27*x)^(5*n+1)/15^(6*n+1). - _Paul D. Hanna_, Jan 24 2008
%F a(n) ~ 3^(-1/2 + 3*n) * (-14 + 5*(5/2)^(6/5))^(1/2 - n) / (2^(3/5) * 5^(9/10) * n^(3/2) * sqrt(Pi)). - _Vaclav Kotesovec_, Nov 28 2017
%e A(x) = 1 + 3*x + 15*x^2 + 210*x^3 + 3510*x^4 + 65562*x^5 +...
%e A(x)^6 = 1 + 18*x + 225*x^2 + 3150*x^3 + 52650*x^4 + 983430*x^5 +...
%t CoefficientList[1 + InverseSeries[Series[(1+15*x - (1+x)^6)/27, {x, 0, 20}], x], x] (* _Vaclav Kotesovec_, Nov 28 2017 *)
%o (PARI) {a(n)=local(A=1+3*x+15*x^2+x*O(x^n));for(i=0,n,A=A+(-15*A+14+27*x+A^6)/9);polcoeff(A,n)}
%Y Cf. A120588 - A120600, A120602 - A120607.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 16 2006
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