A178575 Number of permutations of {1,2,...,3n} whose cycle lengths are all divisible by 3.

1, 2, 160, 62720, 68992000, 163235072000, 710399033344000, 5129081020743680000, 57096929922918645760000, 927825111247427993600000000, 21095031729321522862489600000000, 648714415740095471067280179200000000, 26246985260844262759382156050432000000000

Offset

0,2

References

Herbert S. Wilf, Generatingfunctiontology, page 209

Links

Alois P. Heinz, Table of n, a(n) for n = 0..150

H. Crane and P. McCullagh, Reversible Markov structures on divisible set partitions, Journal of Applied Probability, 52(3), 2015.

Formula

a(n) = (-1)^(n/3)*binomial(-1/3,n/3)*n!.

E.g.f.: 1/(1-x^3)^(1/3).

a(n) = ((3*n)!/(n!*3^n))*Product_{i=1..n-1} (1+3*i) (from the Wilf reference).

a(n) ~ (3*n)! / (Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Jun 15 2015

D-finite with recurrence: a(n) = (3*n-1)*(3*n-2)^2*a(n-1), a(0)=1. - Georg Fischer, Jul 02 2021 (from the 3rd formula)

Example

a(1) = 2 because we have (123) and (132).

Maple

a:= n-> factorial(3*n)*(mul(1+3*i, i = 1 .. n-1))/(factorial(n)*3^n): seq(a(n), n = 0 .. 11);

Mathematica

Table[(-1)^(n/3) Binomial[-1/3, n/3]n!, {n, 0, 30, 3}]

Prog

(PARI) v=Vec(serlaplace(1/(1-x^3+O(x^50))^(1/3))); vector(#v\3, n, v[3*n-2])

Crossrefs

Cf. A001818, A258878.

Sequence in context: A378781 A179958 A272244 * A069580 A332218 A202107

Adjacent sequences: A178572 A178573 A178574 * A178576 A178577 A178578

Keyword

nonn

Author

Geoffrey Critzer, Dec 23 2010

Status

approved