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A004991
a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k + 4).
5
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
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
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