A246610 Number of endofunctions on [n] whose cycle lengths are multiples of 3.

1, 0, 0, 2, 24, 300, 4480, 78750, 1591296, 36355256, 927244800, 26127386010, 806251494400, 27046291980708, 980094896062464, 38158333538165750, 1588601646620835840, 70427042234715548400, 3312574102411273437184, 164767312911755127462066, 8641342923227371929600000

Offset: 0

Author

Alois P. Heinz, Aug 31 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

Column k=3 of A246609.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+3)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= a->add(b(j, 3)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);

Mathematica

CoefficientList[Series[1/(1+LambertW[-x]^3)^(1/3),{x,0,20}],x] * Range[0,20]!  (* Vaclav Kotesovec, Sep 01 2014 *)

Formula

E.g.f.: 1/(1+LambertW(-x)^3)^(1/3). - Vaclav Kotesovec, Sep 01 2014

a(n) ~ Gamma(5/6) * (n^(n-1/3) / (12^(1/3) * sqrt(Pi))) * (1 - 2^(7/6) * Gamma(1/3)^3 / (9 * Pi^(3/2) * sqrt(n))). - Vaclav Kotesovec, Sep 01 2014