A047657 Sextuple factorial numbers: a(n) = Product_{k=0..n-1} (6*k+2).

1, 2, 16, 224, 4480, 116480, 3727360, 141639680, 6232145920, 311607296000, 17450008576000, 1081900531712000, 73569236156416000, 5444123475574784000, 435529878045982720000, 37455569511954513920000, 3445912395099815280640000, 337699414719781897502720000

Offset

0,2

Links

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

Formula

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

a(n) = 2^n*A007559(n).

a(n) = A084941(n)/A000142(n)*A000079(n) = 6^n*Pochhammer(1/3, n) = 1/2*6^n*Gamma(n+1/3)*sqrt(3)*Gamma(2/3)/Pi. - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Let b(n) = b(n-1) + 6; then a(n) = b(n)*a(n-1). - Roger L. Bagula, Sep 17 2008

G.f.: 1/(1-2*x/(1-6*x/(1-8*x/(1-12*x/(1-14*x/(1-18*x/(1-20*x/(1-24*x/(1-26*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012

a(n) = (-4)^n*Sum_{k=0..n} (3/2)^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/G(0) where G(k) = 1 - x*(6*k+2)/( 1 - 6*x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 23 2013

D-finite with recurrence: a(n) +2*(-3*n+2)*a(n-1)=0. - R. J. Mathar, Jan 17 2020

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

Maple

a:= n->product(6*j+2, j=0..n-1); seq(a(n), n=0..20); # G. C. Greubel, Aug 18 2019

Mathematica

b[1]=2; b[n_]:= b[n] = b[n-1] +6; a[0]=1; a[1]=2; a[n_]:= a[n] = a[n-1]*b[n]; Table[a[n], {n, 0, 20}] (* Roger L. Bagula, Sep 17 2008 *)
FoldList[Times, 1, 6*Range[0, 20]+2] (* Harvey P. Dale, Aug 06 2013 *)
Table[6^n*Pochhammer[1/3, n], {n, 0, 20}] (* G. C. Greubel, Aug 18 2019 *)

Prog

(PARI) vector(20, n, n--; prod(k=0, n-1, 6*k+2)) \\ G. C. Greubel, Aug 18 2019
(Magma) [1] cat [(&*[6*k+2: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 18 2019
(Sage) [product(6*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 18 2019
(GAP) List([0..20], n-> Product([0..n-1], k-> 6*k+2) ); # G. C. Greubel, Aug 18 2019

Crossrefs

Cf. A007559, A008542, A011781.

Cf. A000165, A008544, A001813, A047055, A084947, A084948, A084949.

Sequence in context: A188688 A188844 A187657 * A233141 A223631 A188500

Adjacent sequences: A047654 A047655 A047656 * A047658 A047659 A047660

Keyword

nonn,easy

Author

Joe Keane (jgk(AT)jgk.org)

Status

approved