A246613 Number of endofunctions on [n] whose cycle lengths are multiples of 6.

1, 0, 0, 0, 0, 0, 120, 5040, 161280, 4898880, 151200000, 4870182240, 165589522560, 5970883913280, 228632817853440, 9292843409850000, 400385970127503360, 18252778818295082880, 878619947953407528960, 44562094805330883023040, 2376296054962478899200000

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=6 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+6)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= a->add(b(j, 6)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);

Mathematica

CoefficientList[Series[1/(1-LambertW[-x]^6)^(1/6),{x,0,20}],x] * Range[0,20]!  (* Vaclav Kotesovec, Sep 01 2014 *)

Formula

E.g.f.: 1/(1-LambertW(-x)^6)^(1/6). - Vaclav Kotesovec, Sep 01 2014

a(n) ~ n^(n-5/12) * 2^(1/4) * (sqrt(Pi) / (3^(1/6) * Gamma(1/12))) * (1 - 85 * sqrt(2/n) * Gamma(1/12) / (432 * Gamma(7/12))). - Vaclav Kotesovec, Sep 01 2014