A034164 Related to triple factorial numbers 2*A034000(n+1).

1, 5, 30, 198, 1386, 10098, 75735, 580635, 4528953, 35819901, 286559208, 2314516680, 18846778680, 154543585176, 1274984577702, 10574872085646, 88123934047050, 737458184920050, 6194648753328420, 52212039492339540, 441429061162507020, 3742550735942994300

Offset: 0

Author

Wolfdieter Lang

Keywords

• nonn
• easy

Links

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

Crossrefs

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

Programs

Magma

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

Maple

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

Mathematica

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

Sage

def A034164_list(prec):
    P. = 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

Formula

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