0,2

Vincenzo Librandi, Table of n, a(n) for n = 0..200

a(n) = 3^n*(3*n+2)!!!/(n+2)!, where (3*n+2)!!! = 2*A034000(n+1).

G.f.: (1 - 3*x - (1-9*x)^(1/3))/(3*x)^2.

G.f.: 2F1( (1, 5/3); 3; 9 x ). - Olivier Gérard, Feb 15 2011

D-finite with recurrence: (n+2)*a(n) - 3*(3*n+2)*a(n-1) = 0. - R. J. Mathar, Oct 29 2012

a(n) = 3^(2*n+1) * Gamma(n+5/3) / ((n+2) * Gamma(2/3) * Gamma(n+2)). - Vaclav Kotesovec, Feb 09 2014

Integral representation as the n-th moment of a positive function on (0,9): a(n) = Integral_{x=0..9} x^n*W(x) dx, n >= 0, where W(x) = (1/18)*9^(1/3)*sqrt(3)*x^(2/3)*(1-x/9)^(1/3)/Pi. This representation is unique as W(x) is the solution of the Hausdorff moment problem. - Karol A. Penson, Nov 07 2015

Sum_{n>=0} 1/a(n) = 15/16 + (27/64)*(Pi*sqrt(3)/3 - log(3)). - Amiram Eldar, Dec 02 2022

a(n) ~ 3^(2*n+1) * n^(-4/3) / Gamma(2/3). - Amiram Eldar, Aug 19 2025

seq(coeff(series((1-3*x-(1-9*x)^(1/3))/(3*x)^2, x, n+2), x, n), n = 0..32); # G. C. Greubel, Sep 17 2019

CoefficientList[Series[ HypergeometricPFQ[{1, 5/3}, {3}, 9 x], {x, 0, 20}], x]

Table[FullSimplify[3^(2*n+1) * Gamma[n+5/3] / ((n+2) * Gamma[2/3] * Gamma[n+2])], {n, 0, 20}] (* Vaclav Kotesovec, Feb 09 2014 *)

CoefficientList[Series[(1 -3x -(1-9 x)^(1/3))/(3 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 10 2014 *)

(PARI) my(x='x+O('x^30)); Vec((1 -3*x -(1-9*x)^(1/3))/(3*x)^2) \\ G. C. Greubel, Sep 17 2019

(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1 -3*x -(1-9*x)^(1/3))/(3*x)^2 )); // G. C. Greubel, Sep 17 2019

(SageMath)

def A034164_list(prec):

P.<x> = PowerSeriesRing(QQ, prec)

return P((1 -3*x -(1-9*x)^(1/3))/(3*x)^2).list()

A034164_list(30) # G. C. Greubel, Sep 17 2019

Cf. A004990, A025748, A034000, A073006, A185047.

Sequence in context: A196471 A265279 A386835 * A322257 A103433 A081015

Adjacent sequences: A034161 A034162 A034163 * A034165 A034166 A034167

nonn,easy

approved