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%I #7 Nov 28 2017 04:12:42
%S 1,3,15,270,5505,124818,3028200,76896180,2018211930,54311811330,
%T 1490518569747,41556060361920,1173726329836125,33513124885393020,
%U 965755118941566180,28051840723006217040,820439774630057541690
%N G.f. satisfies: 37*A(x) = 36 + 81*x + A(x)^10, 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+37*x - (1+x)^10)/81). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(9*n,n)/(8*n+1) * (36+81*x)^(8*n+1)/37^(9*n+1). - _Paul D. Hanna_, Jan 24 2008
%F a(n) ~ 3^(-1 + 4*n) * (-36 + 9*(37/10)^(10/9))^(1/2 - n) / (2^(5/9) * 5^(1/18) * 37^(4/9) * n^(3/2) * sqrt(Pi)). - _Vaclav Kotesovec_, Nov 28 2017
%e A(x) = 1 + 3*x + 15*x^2 + 270*x^3 + 5505*x^4 + 124818*x^5 +...
%e A(x)^10 = 1 + 30*x + 555*x^2 + 9990*x^3 + 203685*x^4 + 4618266*x^5 +...
%t CoefficientList[1 + InverseSeries[Series[(1+37*x - (1+x)^10)/81, {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+(-37*A+36+81*x+A^10)/27);polcoeff(A,n)}
%Y Cf. A120588 - A120606.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 16 2006
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