A049308 Sextuple factorial numbers: Product_{k=0..n-1} (6*k+4).

1, 4, 40, 640, 14080, 394240, 13404160, 536166400, 24663654400, 1282510028800, 74385581670400, 4760677226905600, 333247405883392000, 25326802847137792000, 2076797833465298944000, 182758209344946307072000, 17179271678424952864768000, 1717927167842495286476800000

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..340

Formula

E.g.f.: (1-6*x)^(-2/3).

a(n) = 2^n*A008544(n).

G.f.: 1/(1-4*x/(1-6*x/(1-10*x/(1-12*x/(1-16*x/(1-18*x/(1-22*x/(1-24*x/(1-28*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012

a(n) = (-2)^n*Sum_{k=0..n} 3^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012

G.f.: ( 1 - 1/Q(0) )/x/2 where Q(k) = 1 - x*(6*k-2)/(1 - x*(6*k+6)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013

D-finite with recurrence: a(n) = 2*(3*n-1)*a(n-1). - R. J. Mathar, Jan 17 2020

From G. C. Greubel, Mar 29 2022: (Start)

a(n) = 6^n * Pochhammer(n, 2/3).

G.f.: Hypergeometric2F0([1, 2/3], [], 6*x). (End)

Sum_{n>=0} 1/a(n) = 1 + exp(1/6)*(Gamma(2/3) - Gamma(2/3, 1/6))/6^(1/3). - Amiram Eldar, Dec 18 2022

Mathematica

Table[6^n*Pochhammer[2/3, n], {n, 0, 30}] (* G. C. Greubel, Mar 29 2022 *)

Prog

(Magma) [n le 2 select 4^(n-1) else 2*(3*n-1)*Self(n-1): n in [1..30]]; // G. C. Greubel, Mar 29 2022
(Sage) [6^n*rising_factorial(2/3, n) for n in (0..30)] # G. C. Greubel, Mar 29 2022
(PARI) a(n) = prod(k=0, n-1, 6*k+4); \\ Michel Marcus, Mar 30 2022

Crossrefs

Cf. A008543, A008544, A011781, A048994.

Sequence in context: A365570 A365587 A090359 * A227055 A343446 A205671

Adjacent sequences: A049305 A049306 A049307 * A049309 A049310 A049311

Keyword

nonn,easy

Author

Joe Keane (jgk(AT)jgk.org)

Status

approved