A102736 Number of permutations of n elements without cycles whose length is a multiple of 3.

1, 1, 2, 4, 16, 80, 400, 2800, 22400, 179200, 1792000, 19712000, 216832000, 2818816000, 39463424000, 552487936000, 8839806976000, 150276718592000, 2554704216064000, 48539380105216000, 970787602104320000, 19415752042086400000, 427146544925900800000, 9824370533295718400000, 225960522265801523200000, 5649013056645038080000000, 146874339472770990080000000, 3818732826292045742080000000

Offset

0,3

Comments

Differs from A247007 first at n=27. - Alois P. Heinz, Sep 09 2014

Links

Alois P. Heinz, Table of n, a(n) for n = 0..450

Formula

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

a(n) ~ n! * 3^(1/3) / (GAMMA(2/3) * n^(1/3)). - Vaclav Kotesovec, Mar 15 2014

Example

G.f. = 1 + x + 2*x^2 + 4*x^3 + 16*x^4 + 80*x^5 + 400*x^6 + 2800*x^7 + ...

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
      irem(j, 3)=0, 0, a(n-j)*(j-1)!*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 31 2017

Mathematica

nn=21; a=Sum[x^n/n, {n, 3, nn, 3}]; Range[0, nn]!CoefficientList[Series[Exp[Log[1/(1-x)]-a], {x, 0, nn}], x]  (* Geoffrey Critzer, Nov 11 2012 *)
a[ n_] := If[ n < 0, 0, n! With[{m = Quotient[n, 3]}, (-1)^m Binomial[-2/3, m]]]; (* Michael Somos, Aug 05 2016 *)

Prog

(PARI) {a(n) = my(m); if( n<0, 0, m = n\3; n! * (-1)^m * binomial(-2/3, m))}; /* Michael Somos, Aug 05 2016 */

Crossrefs

Cf. A000090, A000246.

Sequence in context: A058926 A340900 A347631 * A247007 A103619 A027436

Adjacent sequences: A102733 A102734 A102735 * A102737 A102738 A102739

Keyword

easy,nonn

Author

Vladeta Jovovic, Feb 08 2005

Status

approved