A092618 -id:A092618 - cp's OEIS frontend

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

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

Offset: 0

Joe Keane (jgk(AT)jgk.org)

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

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.

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

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

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

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

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

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

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)

Name changed by Arkadiusz Wesolowski, Oct 04 2011

A029572 Number of permutations of an n-set containing a 5-cycle.

Original entry on oeis.org

0, 0, 0, 0, 0, 24, 144, 1008, 8064, 72576, 653184, 7185024, 86220288, 1120863744, 15692092416, 237124952064, 3793999233024, 64497986961408, 1160963765305344, 22058311540801536, 441004037348818944, 9261084784325197824, 203743865255154352128, 4686108900868550098944

Offset: 0

Rob Pratt

nonn

Column k=5 of A293211.

nn = 20; a = Log[1/(1 - x)] - x^5/5; Range[0, nn]! CoefficientList[Series[1/(1 - x) - Exp[a], {x, 0, nn}],x] (* Geoffrey Critzer, Jun 01 2013 *)

my(x='x+O('x^66)); concat([0,0,0,0,0], Vec(serlaplace((1-exp(-x^5/5))/(1-x)))) \\ Joerg Arndt, Jun 01 2013

a(n) = n!*(1 - Sum_{k=0..floor(n/5)} (-1)^k/(k!*5^k)).
a(n)/n! is asymptotic to 1-e^(-1/5) = 1 - A092618.
E.g.f.: (1-exp(-x^5/5))/(1-x) - Geoffrey Critzer, Jun 01 2013

cp's OEIS Frontend

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

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