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A120593 G.f. satisfies: 5*A(x) = 4 + x + A(x)^4, starting with [1,1,6]. 3
1, 1, 6, 76, 1201, 21252, 402892, 8001412, 164321982, 3461110532, 74358814838, 1623152780808, 35897318940028, 802620009567628, 18112759482614328, 412020809942451504, 9437537418826749369, 217486633306640519124 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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. satisfies:

(1) A(x) = 1 + Series_Reversion(1+5*x - (1+x)^4).

(2) A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (4+x)^(3*n+1)/5^(4*n+1), by Lagrange Inversion.

(3) A(x) = F(x/A(x)) and F(x) = A(x*F(x)) where F(x) = (4 + F(x)^4)/(5-x)  is the g.f. of A244856. - Paul D. Hanna, Jul 09 2014

a(n) ~ 2^(-7/3 + 3*n) * (-32 + 15*10^(1/3))^(1/2 - n) / (5^(1/3) * n^(3/2) * sqrt(3*Pi)). - Vaclav Kotesovec, Nov 28 2017

EXAMPLE

A(x) = 1 + x + 6*x^2 + 76*x^3 + 1201*x^4 + 21252*x^5 +...

A(x)^4 = 1 + 4*x + 30*x^2 + 380*x^3 + 6005*x^4 + 106260*x^5 +...

MATHEMATICA

CoefficientList[1 + InverseSeries[Series[1+5*x - (1+x)^4, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)

PROG

(PARI) {a(n)=local(A=1+x+6*x^2+x*O(x^n)); for(i=0, n, A=A-5*A+4+x+A^4); polcoeff(A, n)}

CROSSREFS

Cf. A120588 - A120592, A120594 - A120607; A244856.

Sequence in context: A066797 A333097 A231281 * A318614 A236954 A116874

Adjacent sequences:  A120590 A120591 A120592 * A120594 A120595 A120596

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 16 2006, Jan 24 2008

STATUS

approved

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Last modified September 27 03:05 EDT 2021. Contains 347673 sequences. (Running on oeis4.)