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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 (list; graph; refs; listen; history; text; internal format)
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 3-cycles.

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)my-deja.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).

D-finite with recurrence a(n) = (3*n-1)*(3*n-2)*a(n-1) 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

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Last modified December 2 18:27 EST 2022. Contains 358510 sequences. (Running on oeis4.)