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A004991 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k + 4).

%I A004991 #28 Dec 02 2022 07:03:58
%S A004991 1,12,126,1260,12285,117936,1120392,10563696,99034650,924323400,
%T A004991 8596207620,79710288840,737320171770,6806032354800,62712726697800,
%U A004991 576957085619760,5300793224131545,48642573115560060,445890253559300550,4083416006279910300,37363256457461179245,341606916182502210240
%N A004991 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k + 4).
%H A004991 G. C. Greubel, <a href="/A004991/b004991.txt">Table of n, a(n) for n = 0..1000</a>
%F A004991 G.f.: (1 - 9*x)^(-4/3).
%F A004991 a(n) ~ 3*Gamma(1/3)^-1*n^(1/3)*3^(2*n)*(1 + 2/9*n^-1 - ...).
%F A004991 a(n) = (3^(2*n))/(Integral_{x=0..1} (1-x^3)^n dx). - Al Hakanson (hawkuu(AT)excite.com), Dec 04 2003
%F A004991 D-finite with recurrence: n*a(n) +3*(-3*n-1)*a(n-1)=0. - _R. J. Mathar_, Jan 17 2020
%F A004991 Sum_{n>=0} 1/a(n) = sqrt(3)*Pi/8 + 3*log(3)/8. - _Amiram Eldar_, Dec 02 2022
%p A004991 a:= n-> (3^n/n!)*product(3*k+4, k=0..n-1); seq(a(n), n=0..25); # _G. C. Greubel_, Aug 22 2019
%t A004991 Table[9^n*Pochhammer[4/3, n]/n!, {n,0,25}] (* _G. C. Greubel_, Aug 22 2019 *)
%t A004991 Table[3^n/n! Product[3k+4,{k,0,n-1}],{n,0,30}] (* or *) CoefficientList[ Series[ 1/Surd[(1-9x)^4,3],{x,0,30}],x] (* _Harvey P. Dale_, Aug 02 2021 *)
%o A004991 (PARI) a(n) = 3^n*prod(k=0,n-1, 3*k+4)/n!;
%o A004991 vector(25, n, n--; a(n)) \\ _G. C. Greubel_, Aug 22 2019
%o A004991 (Magma) [1] cat [3^n*(&*[3*k+4: k in [0..n-1]])/Factorial(n): n in [1..25]]; // _G. C. Greubel_, Aug 22 2019
%o A004991 (Sage) [9^n*rising_factorial(4/3, n)/factorial(n) for n in (0..25)] # _G. C. Greubel_, Aug 22 2019
%o A004991 (GAP) List([0..25], n-> 3^n*Product([0..n-1], k-> 3*k+4)/Factorial(n) ); # _G. C. Greubel_, Aug 22 2019
%Y A004991 Cf. A004988, A004989, A004990, A004992.
%K A004991 nonn,easy
%O A004991 0,2
%A A004991 Joe Keane (jgk(AT)jgk.org)
%E A004991 Terms a(16) onward added by _G. C. Greubel_, Aug 22 2019