A120604 G.f. satisfies: 24*A(x) = 23 + 64*x + A(x)^8, starting with [1,4,28].

1, 4, 28, 616, 15820, 453208, 13894552, 445970128, 14796844588, 503423385080, 17467725995720, 615756709476272, 21990183407958584, 793912445913712496, 28928560840589374640, 1062498482335560005024, 39293868860176487815916

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..16.

Formula

G.f.: A(x) = 1 + Series_Reversion((1+24*x - (1+x)^8)/64). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(8*n,n)/(7*n+1) * (23+64*x)^(7*n+1)/24^(8*n+1). - Paul D. Hanna, Jan 24 2008

a(n) ~ 4^(-1 + 3*n) * (-23 + 21*3^(1/7))^(1/2 - n) / (3^(3/7) * n^(3/2) * sqrt(7*Pi)). - Vaclav Kotesovec, Nov 28 2017

Example

A(x) = 1 + 4*x + 28*x^2 + 616*x^3 + 15820*x^4 + 453208*x^5 +...
A(x)^8 = 1 + 32*x + 672*x^2 + 14784*x^3 + 379680*x^4 + 10876992*x^5 +...

Mathematica

CoefficientList[1 + InverseSeries[Series[(1+24*x - (1+x)^8)/64, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)

Prog

(PARI) {a(n)=local(A=1+4*x+28*x^2+x*O(x^n)); for(i=0, n, A=A+(-24*A+23+64*x+A^8)/16); polcoeff(A, n)}

Crossrefs

Cf. A120588 - A120603, A120605 - A120607.

Sequence in context: A134048 A091969 A101346 * A203279 A081792 A336622

Adjacent sequences: A120601 A120602 A120603 * A120605 A120606 A120607

Keyword

nonn

Author

Paul D. Hanna, Jun 16 2006

Status

approved