

A052502


Number of permutations sigma of [3n] without fixed points such that sigma^3 = Id.


17



1, 2, 40, 2240, 246400, 44844800, 12197785600, 4635158528000, 2345390215168000, 1524503639859200000, 1237896955565670400000, 1227993779921145036800000, 1461312598106162593792000000, 2054605512937264606871552000000
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OFFSET

0,2


COMMENTS

For n >= 1 a(n) is the size of the conjugacy class in the symmetric group S_(3n) consisting of permutations whose cycle decomposition is a product of n disjoint 3cycles.


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..210
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 27


FORMULA

From Ahmed Fares (ahmedfares(AT)mydeja.com), Apr 21 2001: (Start)
a(n) = (3*n)!/(3^n * n!).
a(n) ~ sqrt(3) * 9^n * (n/e)^(2n). (End)
E.g.f.: (every third coefficient of) exp(x^3/3).
G.f.: hypergeometric3F0([1/3, 2/3, 1], [], 9*x).
Dfinite with recurrence a(n) = (3*n1)*(3*n2)*a(n1) for n >= 1, with a(0) = 1.
Write the generating function for this sequence in the form A(x) = sum {n >= 0} a(n)* x^(2*n+1)/(2*n+1)!. The g.f. A(x) satisfies A'(x)*( 1  A(x)^2) = 1. Robert Israel remarks that consequently A(x) is a root of z^3  3*z + 3*x with A(0) = 0. Cf. A001147, A052504 and A060706.  Peter Bala, Jan 02 2015


MAPLE

spec := [S, {S=Set(Union(Cycle(Z, card=3)))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);


MATHEMATICA

Table[(3*n)!/(3^n*n!), {n, 0, 20}] (* G. C. Greubel, May 14 2019 *)


PROG

(PARI) {a(n) = (3*n)!/(3^n*n!)}; \\ G. C. Greubel, May 14 2019
(Magma) [Factorial(3*n)/(3^n*Factorial(n)): n in [0..20]]; // G. C. Greubel, May 14 2019
(Sage) [factorial(3*n)/(3^n*factorial(n)) for n in (0..20)] # G. C. Greubel, May 14 2019
(GAP) List([0..20], n> Factorial(3*n)/(3^n*Factorial(n))) # G. C. Greubel, May 14 2019


CROSSREFS

Cf. A000142. Row sums of triangle A060063.
First column of array A091752 (also negative of second column).
Equals row sums of A157702.  Johannes W. Meijer, Mar 07 2009
Karol A. Penson suggested that the row sums of A060063 coincide with this entry.
Cf. A001147, A052504, A060706, A261317, A261381.
Trisection of column k=3 of A261430.
Sequence in context: A000816 A000819 A060079 * A209289 A246742 A293950
Adjacent sequences: A052499 A052500 A052501 * A052503 A052504 A052505


KEYWORD

easy,nonn


AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000


EXTENSIONS

Edited by Wolfdieter Lang, Feb 13 2004
Title improved by Geoffrey Critzer, Aug 14 2015


STATUS

approved



