A120594 G.f. satisfies: 8*A(x) = 7 + 8*x + A(x)^4, starting with [1,2,6].

1, 2, 6, 44, 394, 3948, 42364, 476120, 5532714, 65935804, 801461012, 9897836520, 123840983812, 1566487308344, 19999112293944, 257365488659376, 3334967582746218, 43477505482249692, 569854228738577572

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..18.

Formula

G.f.: A(x) = 1 + Series_Reversion((1+8*x - (1+x)^4)/8). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (7+8*x)^(3*n+1)/8^(4*n+1). - Paul D. Hanna, Jan 24 2008

a(n) ~ 2^(-11/6 + 3*n) * (-7 + 6*2^(1/3))^(1/2 - n) / (n^(3/2) * sqrt(3*Pi)). - Vaclav Kotesovec, Nov 28 2017

Example

A(x) = 1 + 2*x + 6*x^2 + 44*x^3 + 394*x^4 + 3948*x^5 + 42364*x^6 +...
A(x)^4 = 1 + 8*x + 48*x^2 + 352*x^3 + 3152*x^4 + 31584*x^5 + 338912*x^6+..

Mathematica

CoefficientList[1 + InverseSeries[Series[(1+8*x - (1+x)^4)/8, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)

Prog

(PARI) {a(n)=local(A=1+2*x+6*x^2+x*O(x^n)); for(i=0, n, A=A+(-8*A+7+8*x+A^4)/4); polcoeff(A, n)}

Crossrefs

Cf. A120588 - A120593, A120595 - A120607.

Sequence in context: A077048 A354309 A277479 * A038180 A058925 A266855

Adjacent sequences: A120591 A120592 A120593 * A120595 A120596 A120597

Keyword

nonn

Author

Paul D. Hanna, Jun 16 2006

Status

approved