A052562 a(n) = 5^n * n!.

1, 5, 50, 750, 15000, 375000, 11250000, 393750000, 15750000000, 708750000000, 35437500000000, 1949062500000000, 116943750000000000, 7601343750000000000, 532094062500000000000, 39907054687500000000000

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• nonn
• easy

Comments

A simple regular expression in a labeled universe.

For n >= 1 a(n) is the order of the wreath product of the symmetric group S_n and the Abelian group (C_5)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 504
• Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Crossrefs

Cf. A000142, A008548, A008546, A034325, A000165, A047055, A047056, A051150, A092514, A092618.

Programs

Magma

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

Maple

spec := [S,{S=Sequence(Union(Z,Z,Z,Z,Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
with(combstruct):A:=[N,{N=Cycle(Union(Z$5))},labeled]: seq(count(A,size=n)/5,n=1..16); # Zerinvary Lajos, Dec 05 2007

Mathematica

Table[5^n*n!, {n, 0, 20}] (* Wesley Ivan Hurt, Sep 28 2013 *)

PARI

{a(n) = 5^n*n!}; \\ G. C. Greubel, May 05 2019

Sage

[5^n*factorial(n) for n in (0..20)] # G. C. Greubel, May 05 2019

Formula

a(n) = A051150(n+1, 0) (first column of triangle).

E.g.f.: 1/(1-5*x).

a(n) = 5*n*a(n-1) with a(0)=1.

G.f.: 1/(1-5*x/(1-5*x/(1-10*x/(1-10*x/(1-15*x/(1-15*x/(1-20*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012

G.f.: 1/Q(0), where Q(k) = 1 - 5*x*(2*k+1) - 25*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 28 2013

a(n) = n!*A000351(n). - R. J. Mathar, Aug 21 2014

From Amiram Eldar, Jun 25 2020: (Start)

Sum_{n>=0} 1/a(n) = e^(1/5) (A092514).

Sum_{n>=0} (-1)^n/a(n) = e^(-1/5) (A092618). (End)

Extensions

Name changed by Arkadiusz Wesolowski, Oct 04 2011