OFFSET
0,2
COMMENTS
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.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..100
Peter Bala, The method of Graves for computing inverse functions.
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 7.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 29.
FORMULA
a(n) = (5n)! * [x^(5n)] exp(x^5/5).
From Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001: (Start)
a(n) = (5*n)! / (n! * 5^n).
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).
a(n) ~ sqrt(5) * 625^n * (n/e)^(4n). (End)
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
A(x) = Series_Reversion(x - x^5/5); A(x)^5 - 5*A(x) + 5*x = 0. - Peter Bala, Feb 22 2026
MAPLE
spec := [S, {S=Set(Union(Cycle(Z, card=5)))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
# uses the method of Graves to compute the k-th powers of an inverse function
d := proc (n, x) option remember; if n = 0 then x^k else simplify( 1/(1 - x^4)*diff(d(n-1, x), x) ) end if end proc:
k := 1:
seq( eval(d(4*n+1, x), x = 0), n = 0..10); # Peter Bala, Feb 22 2026
MATHEMATICA
nn = 50; Select[Range[0, nn]! CoefficientList[Series[Exp[x^5/5], {x, 0, nn}], x], # > 0 &] (* Geoffrey Critzer, Aug 19 2012 *)
PROG
(PARI) {a(n) = (5*n)!/(5^n*n!)}; \\ G. C. Greubel, May 14 2019
(Magma) [Factorial(5*n)/(5^n*Factorial(n)): n in [0..15]]; // G. C. Greubel, May 14 2019
(SageMath) [factorial(5*n)/(5^n*factorial(n)) for n in (0..15)] # G. C. Greubel, May 14 2019
(GAP) List([0..15], n-> Factorial(5*n)/(5^n*Factorial(n))); # G. C. Greubel, May 14 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
INRIA Encyclopedia of Combinatorial Structures, Jan 25 2000
STATUS
approved