A004988 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k + 2).

1, 6, 45, 360, 2970, 24948, 212058, 1817640, 15677145, 135868590, 1182056733, 10316131488, 90266150520, 791564704560, 6954461332920, 61199259729696, 539318476367946, 4758692438540700, 42035116540442850, 371678925199705200, 3289358488017391020

Offset

0,2

Links

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

Formula

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

a(n) = 9^n*Gegenbauer_C(n,1/3,1). - Paul Barry, Apr 21 2009

a(n) = Product_{k=1..n} (9 - 3/k). - Michel Lagneau, Sep 16 2012

a(n) = (-9)^n*binomial(-2/3, n). - R. J. Mathar, Sep 16 2012

D-finite with recurrence: n*a(n) +3*(-3*n+1)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012

a(n) = 9^n * Gamma(n+2/3) / (Gamma(2/3) * Gamma(n+1)). - Vaclav Kotesovec, Feb 09 2014

Sum_{n>=0} 1/a(n) = 9/8 + sqrt(3)*Pi/32 - 3*log(3)/32. - Amiram Eldar, Dec 02 2022

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) = sqrt(3)/(2*Pi) * 1/(x*(9 - x)^2)^(1/3). The weight function w(x) is the solution of the Hausdorff moment problem on (0, 9) with moments equal to a(n). As a consequence this representation is unique. Cf. A004987. - Peter Bala, Oct 13 2024

Maple

A004988 := proc(n)
        binomial(-2/3, n)*(-9)^n ;
end proc: # R. J. Mathar, Sep 16 2012

Mathematica

Table[FullSimplify[9^n*Gamma[n+2/3]/(Gamma[2/3]*Gamma[n+1])], {n, 0, 20}] (* Vaclav Kotesovec, Feb 09 2014 *)
CoefficientList[Series[(1-9x)^(-2/3), {x, 0, 20}], x] (* Vincenzo Librandi, Feb 10 2014 *)
Table[9^n*Pochhammer[2/3, n]/n!, {n, 0, 20}] (* G. C. Greubel, Aug 22 2019 *)

Prog

(PARI) a(n)=if(n<0, 0, prod(k=0, n-1, 3*k+2)*3^n/n!)
(Magma) [1] cat [3^n*&*[3*k+2: k in [0..n-1]]/Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 22 2019
(Sage) [9^n*rising_factorial(2/3, n)/factorial(n) for n in (0..20)] # G. C. Greubel, Aug 22 2019
(GAP) List([0..20], n-> 3^n*Product([0..n-1], k-> 3*k+2)/Factorial(n) ); # G. C. Greubel, Aug 22 2019

Crossrefs

Cf. A004987, A004989, A004990, A004991, A004992.

Sequence in context: A097129 A048663 A095003 * A257633 A371779 A007193

Adjacent sequences: A004985 A004986 A004987 * A004989 A004990 A004991

Keyword

nonn,easy

Author

Joe Keane (jgk(AT)jgk.org)

Status

approved