A052562 - OEIS

%I #58 Aug 12 2024 13:23:55

%S 1,5,50,750,15000,375000,11250000,393750000,15750000000,708750000000,

%T 35437500000000,1949062500000000,116943750000000000,

%U 7601343750000000000,532094062500000000000,39907054687500000000000

%N a(n) = 5^n * n!.

%C A simple regular expression in a labeled universe.

%C 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

%H Vincenzo Librandi, <a href="/A052562/b052562.txt">Table of n, a(n) for n = 0..300</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=504">Encyclopedia of Combinatorial Structures 504</a>

%H Michael Z. Spivey and Laura L. Steil, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

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

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

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

%F 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

%F 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

%F a(n) = n!*A000351(n). - _R. J. Mathar_, Aug 21 2014

%F From _Amiram Eldar_, Jun 25 2020: (Start)

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

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

%p spec := [S,{S=Sequence(Union(Z,Z,Z,Z,Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%p with(combstruct):A:=[N,{N=Cycle(Union(Z$5))},labeled]: seq(count(A,size=n)/5,n=1..16); # _Zerinvary Lajos_, Dec 05 2007

%t Table[5^n*n!, {n, 0, 20}] (* _Wesley Ivan Hurt_, Sep 28 2013 *)

%o (Magma)[5^n*Factorial(n): n in [0..20]]; // _Vincenzo Librandi_, Oct 05 2011

%o (PARI) {a(n) = 5^n*n!}; \\ _G. C. Greubel_, May 05 2019

%o (Sage) [5^n*factorial(n) for n in (0..20)] # _G. C. Greubel_, May 05 2019

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

%K nonn,easy

%O 0,2

%A Joe Keane (jgk(AT)jgk.org)

%E Name changed by _Arkadiusz Wesolowski_, Oct 04 2011