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

1, 12, 126, 1260, 12285, 117936, 1120392, 10563696, 99034650, 924323400, 8596207620, 79710288840, 737320171770, 6806032354800, 62712726697800, 576957085619760, 5300793224131545, 48642573115560060, 445890253559300550, 4083416006279910300, 37363256457461179245, 341606916182502210240

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

G.f.: (1 - 9*x)^(-4/3).

a(n) ~ 3*Gamma(1/3)^-1*n^(1/3)*3^(2*n)*(1 + 2/9*n^-1 - ...).

a(n) = (3^(2*n))/(Integral_{x=0..1} (1-x^3)^n dx). - Al Hakanson (hawkuu(AT)excite.com), Dec 04 2003

D-finite with recurrence: n*a(n) +3*(-3*n-1)*a(n-1)=0. - R. J. Mathar, Jan 17 2020

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

Maple

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

Mathematica

Table[9^n*Pochhammer[4/3, n]/n!, {n, 0, 25}] (* G. C. Greubel, Aug 22 2019 *)
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 *)

Prog

(PARI) a(n) = 3^n*prod(k=0, n-1, 3*k+4)/n!;
vector(25, n, n--; a(n)) \\ G. C. Greubel, Aug 22 2019
(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
(Sage) [9^n*rising_factorial(4/3, n)/factorial(n) for n in (0..25)] # G. C. Greubel, Aug 22 2019
(GAP) List([0..25], n-> 3^n*Product([0..n-1], k-> 3*k+4)/Factorial(n) ); # G. C. Greubel, Aug 22 2019

Crossrefs

Cf. A004988, A004989, A004990, A004992.

Sequence in context: A343769 A348463 A159736 * A228008 A101602 A264896

Adjacent sequences: A004988 A004989 A004990 * A004992 A004993 A004994

Keyword

nonn,easy

Author

Joe Keane (jgk(AT)jgk.org)

Extensions

Terms a(16) onward added by G. C. Greubel, Aug 22 2019

Status

approved