A120606 G.f. satisfies: 36*A(x) = 35 + 81*x + A(x)^9, starting with [1,3,12].

1, 3, 12, 180, 3018, 56238, 1121484, 23406804, 504914175, 11167352013, 251879507880, 5771456609880, 133970974830420, 3143760834627420, 74454455230816008, 1777349666975945784, 42721359085344132657

Offset: 0

Author

Paul D. Hanna, Jun 16 2006

Keywords

• nonn

Comments

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.

Examples

			A(x) = 1 + 3*x + 12*x^2 + 180*x^3 + 3018*x^4 + 56238*x^5 +...
A(x)^9 = 1 + 27*x + 432*x^2 + 6480*x^3 + 108648*x^4 + 2024568*x^5 +...
		

Crossrefs

Cf. A120588 - A120605, A120607.

Programs

Mathematica

CoefficientList[1 + InverseSeries[Series[(1+36*x - (1+x)^9)/81, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)

PARI

{a(n)=local(A=1+3*x+12*x^2+x*O(x^n));for(i=0,n,A=A+(-36*A+35+81*x+A^9)/27);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+36*x - (1+x)^9)/81). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(9*n,n)/(8*n+1) * (35+81*x)^(8*n+1)/36^(9*n+1). - Paul D. Hanna, Jan 24 2008

a(n) ~ 3^(-1 + 4*n) * (-35 + 2^(21/4))^(1/2 - n) / (2^(23/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017