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A120595 G.f. satisfies: 13*A(x) = 12 + 27*x + A(x)^4, starting with [1,3,6]. 3
1, 3, 6, 36, 249, 1932, 16044, 139500, 1253934, 11558316, 108658902, 1037800920, 10041891132, 98230257636, 969814634424, 9651213968784, 96710160474513, 974967422602428, 9881687141571732, 100632995795535588 (list; graph; refs; listen; history; text; internal format)
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..19.

FORMULA

G.f.: A(x) = 1 + Series_Reversion((1+13*x - (1+x)^4)/27).

G.f.: A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (12 + 27*x)^(3*n+1) / 13^(4*n+1). - Paul D. Hanna, Jan 24 2008

a(n) ~ 2^(3*n - 7/3) * 3^(2*n) / (13^(1/3) * sqrt(Pi) * n^(3/2) * (2^(1/3)*13^(4/3) - 32)^(n - 1/2)). - Vaclav Kotesovec, Nov 27 2017

EXAMPLE

A(x) = 1 + 3*x + 6*x^2 + 36*x^3 + 249*x^4 + 1932*x^5 +...

A(x)^4 = 1 + 12*x + 78*x^2 + 468*x^3 + 3237*x^4 + 25116*x^5 +...

MATHEMATICA

CoefficientList[1 + InverseSeries[Series[(1+13*x - (1+x)^4)/27, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 27 2017 *)

PROG

(PARI) {a(n)=local(A=1+3*x+6*x^2+x*O(x^n)); for(i=0, n, A=A+(-13*A+12+27*x+A^4)/9); polcoeff(A, n)}

CROSSREFS

Cf. A120588 - A120594, A120596 - A120607; A245043.

Sequence in context: A003674 A211895 A240986 * A048642 A264702 A077532

Adjacent sequences:  A120592 A120593 A120594 * A120596 A120597 A120598

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 16 2006

STATUS

approved

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Last modified September 16 21:28 EDT 2021. Contains 347473 sequences. (Running on oeis4.)