A034001 One third of triple factorial numbers.

1, 6, 54, 648, 9720, 174960, 3674160, 88179840, 2380855680, 71425670400, 2357047123200, 84853696435200, 3309294160972800, 138990354760857600, 6254565964238592000, 300219166283452416000, 15311177480456073216000

Offset: 1

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..100
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 495.
• Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.
• N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order, Vol. 21 (2004), pp. 83-89; arXiv preprint, arXiv:math/0307064 [math.CO], 2003.

Crossrefs

Cf. A007559, A034000.

Programs

GAP

List([1..20],n->3^(n-1)*Factorial(n)); # Muniru A Asiru, Jul 28 2018

Magma

[3^(n-1)*Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 15 2019

Maple

G(x):=(1-3*x)^(n-3): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od:x:=0:seq(f[n],n=0..16); # Zerinvary Lajos, Apr 04 2009

Mathematica

terms = 17;
CoefficientList[1/(1-3x)^2 + O[x]^terms, x] Range[0, terms-1]! (* Jean-François Alcover, Jul 28 2018 *)
Table[3^(n-1)*n!, {n,20}] (* G. C. Greubel, Aug 15 2019 *)

PARI

vector(20, n, 3^(n-1)*n!) \\ G. C. Greubel, Aug 15 2019

Sage

[3^(n-1)*factorial(n) for n in (1..20)] # G. C. Greubel, Aug 15 2019

Formula

3*a(n) = (3*n)!!! = Product_{j=1..n} 3*j = 3^n*n!.

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

E.g.f.: 1/(1-3*x)^2. - Paul Barry, Sep 14 2004. For offset 0. - Wolfdieter Lang, Apr 06 2017

D-finite with recurrence a(n) - 3*n*a(n-1) = 0. - R. J. Mathar, Dec 02 2012

From Amiram Eldar, Jan 08 2022: (Start)

Sum_{n>=1} 1/a(n) = 3*(exp(1/3)-1).

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(1-exp(-1/3)). (End)