A120597 G.f. satisfies: 9*A(x) = 8 + 8*x + A(x)^5, starting with [1,2,10].

1, 2, 10, 120, 1770, 29208, 516180, 9554640, 182867970, 3589443160, 71861735660, 1461730482160, 30123451315620, 627598216410480, 13197173403868200, 279728425129963680, 5970277970921643570, 128199003794219752920

Offset

0,2

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.

Links

Table of n, a(n) for n=0..17.

Formula

G.f.: A(x) = 1 + Series_Reversion((1+9*x - (1+x)^5)/8). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(5*n,n)/(4*n+1) * (8+8*x)^(4*n+1)/9^(5*n+1). - Paul D. Hanna, Jan 24 2008

a(n) ~ (-1 + 9*sqrt(3)/(10*5^(1/4)))^(1/2 - n) / (3^(3/4) * 5^(1/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

Example

A(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 1770*x^4 + 29208*x^5 +...
A(x)^5 = 1 + 10*x + 90*x^2 + 1080*x^3 + 15930*x^4 + 262872*x^5 +...

Mathematica

CoefficientList[1 + InverseSeries[Series[(1+9*x - (1+x)^5)/8, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)

Prog

(PARI) {a(n)=local(A=1+2*x+10*x^2+x*O(x^n)); for(i=0, n, A=A+(-9*A+8+8*x+A^5)/4); polcoeff(A, n)}

Crossrefs

Cf. A120588 - A120596, A120598 - A120607.

Sequence in context: A110951 A172477 A265942 * A322295 A363586 A256832

Adjacent sequences: A120594 A120595 A120596 * A120598 A120599 A120600

Keyword

nonn

Author

Paul D. Hanna, Jun 16 2006

Status

approved