%I #36 Sep 08 2022 08:44:59
%S 1,24,72576,1743565824,162193467211776,41363226782215962624,
%T 23578031983305871878782976,26242915470187034742010543079424,
%U 51804144968120491069562620291816882176,168779147605615794796420686413626405734580224,858246016274098851318874304509764200194078068965376
%N Number of permutations sigma of [5n] without fixed points such that sigma^5 = Id.
%C For n >= 1 a(n) is the size of the conjugacy class in the symmetric group S_(5n) consisting of permutations whose cycle decomposition is a product of n disjoint 5-cycles.
%H G. C. Greubel, <a href="/A052504/b052504.txt">Table of n, a(n) for n = 0..100</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=29">Encyclopedia of Combinatorial Structures 29</a>
%F a(n) = (5n)! * [x^(5n)] exp(x^5/5).
%F From Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001: (Start)
%F a(n) = (5*n)! / (n! * 5^n).
%F a(0) = 1, a(1) = 24, for n >= 2 a(n) = a(n-1) * C(5*n - 1, 4)* 24 = (5*n-1)*(5*n-2)*(5*n-3)*(5*n-4)*a(n-1).
%F a(n) ~ sqrt(5) * 625^n * (n/e)^(4n). (End)
%F Write the generating function for this sequence in the form A(x) = Sum_{n >= 0} a(n)* x^(4*n+1)/(4*n+1)!. Then A'(x)*( 1 - A(x)^4) = 1. Cf. A052502. - _Peter Bala_, Jan 02 2015
%p spec := [S,{S=Set(Union(Cycle(Z,card=5)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t nn = 50; Select[Range[0, nn]! CoefficientList[Series[Exp[x^5/5], {x, 0, nn}], x], # > 0 &] (* _Geoffrey Critzer_, Aug 19 2012 *)
%o (PARI) {a(n) = (5*n)!/(5^n*n!)}; \\ _G. C. Greubel_, May 14 2019
%o (Magma) [Factorial(5*n)/(5^n*Factorial(n)): n in [0..15]]; // _G. C. Greubel_, May 14 2019
%o (Sage) [factorial(5*n)/(5^n*factorial(n)) for n in (0..15)] # _G. C. Greubel_, May 14 2019
%o (GAP) List([0..15], n-> Factorial(5*n)/(5^n*Factorial(n))) # _G. C. Greubel_, May 14 2019
%Y Cf. A001147, A052502, A060706, A261317, A261381.
%Y Quintisection of column k=5 of A261430.
%K easy,nonn
%O 0,2
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000