A047058 a(n) = 6^n * n!.

1, 6, 72, 1296, 31104, 933120, 33592320, 1410877440, 67722117120, 3656994324480, 219419659468800, 14481697524940800, 1042682221795737600, 81329213300067532800, 6831653917205672755200, 614848852548510547968000

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• nonn
• easy

Comments

For n >= 1, a(n) is the order of the wreath product of the symmetric group S_n and the Abelian group (C_6)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001

a(n) is the number of ways 3 members of each of n different teams can be arranged in a row so that members of the same team are together. - Geoffrey Critzer, Mar 30 2009

From Jianing Song, Mar 29 2021: (Start)

Number of n X n monomial matrices with entries 0, +/-1, +/-w, +/-w^2, where w = (-1 + sqrt(3)*i)/2 is a primitive 3rd root of unity.

a(n) is the order of the group U_n(Z[w]) = {A in M_n(Z[w]): A*A^H = I_n}, the group of n X n unitary matrices over the Eisenstein integers. Here A^H is the conjugate transpose of A. (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 528.

Crossrefs

Cf. A000142, A000165, A008542, A008543, A047053, A051151, A053103, A092515, A092727.

Programs

Magma

[6^n*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 05 2011

Maple

seq( 6^n*n!, n=0..20); # G. C. Greubel, Jun 08 2020

Mathematica

Table[6^n n!,{n,0,20}] (* Harvey P. Dale, Mar 30 2018 *)

Formula

a(n) = A051151(n+1, 0).

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

G.f.: 1/(1 -6*x/(1 - 6*x/(1 - 12*x/(1 - 12*x/(1 - 18*x/(1 - 18*x/(1 - 24*x/(1 - 24*x/(1 - 30*x/(1 - 30*x/(1 -... (continued fraction). - Philippe Deléham, Jan 08 2012

From Amiram Eldar, Jun 25 2020: (Start)

Sum_{n>=0} 1/a(n) = e^(1/6) (A092515).

Sum_{n>=0} (-1)^n/a(n) = e^(-1/6) (A092727). (End)

Extensions

Name changed by Arkadiusz Wesolowski, Oct 04 2011