%I #53 Dec 02 2022 07:04:17

%S 1,6,45,360,2970,24948,212058,1817640,15677145,135868590,1182056733,

%T 10316131488,90266150520,791564704560,6954461332920,61199259729696,

%U 539318476367946,4758692438540700,42035116540442850,371678925199705200,3289358488017391020

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

%H Vincenzo Librandi, <a href="/A004988/b004988.txt">Table of n, a(n) for n = 0..200</a>

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

%F a(n) = 9^n*Gegenbauer_C(n,1/3,1). - _Paul Barry_, Apr 21 2009

%F a(n) = Product_{k=1..n} (9 - 3/k). - _Michel Lagneau_, Sep 16 2012

%F a(n) = (-9)^n*binomial(-2/3, n). - _R. J. Mathar_, Sep 16 2012

%F D-finite with recurrence: n*a(n) +3*(-3*n+1)*a(n-1) = 0. - _R. J. Mathar_, Dec 03 2012

%F a(n) = 9^n * Gamma(n+2/3) / (Gamma(2/3) * Gamma(n+1)). - _Vaclav Kotesovec_, Feb 09 2014

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

%p A004988 := proc(n)

%p binomial(-2/3,n)*(-9)^n ;

%p end proc: # _R. J. Mathar_, Sep 16 2012

%t Table[FullSimplify[9^n*Gamma[n+2/3]/(Gamma[2/3]*Gamma[n+1])],{n,0,20}] (* _Vaclav Kotesovec_, Feb 09 2014 *)

%t CoefficientList[Series[(1-9x)^(-2/3), {x, 0, 20}], x] (* _Vincenzo Librandi_, Feb 10 2014 *)

%t Table[9^n*Pochhammer[2/3, n]/n!, {n,0,20}] (* _G. C. Greubel_, Aug 22 2019 *)

%o (PARI) a(n)=if(n<0,0,prod(k=0,n-1,3*k+2)*3^n/n!)

%o (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

%o (Sage) [9^n*rising_factorial(2/3, n)/factorial(n) for n in (0..20)] # _G. C. Greubel_, Aug 22 2019

%o (GAP) List([0..20], n-> 3^n*Product([0..n-1], k-> 3*k+2)/Factorial(n) ); # _G. C. Greubel_, Aug 22 2019

%Y Cf. A004989, A004990, A004991, A004992.

%K nonn,easy

%O 0,2

%A Joe Keane (jgk(AT)jgk.org)