A246616 Number of endofunctions on [n] whose cycle lengths are multiples of 9.

1, 0, 0, 0, 0, 0, 0, 0, 0, 40320, 3628800, 219542400, 11496038400, 570031862400, 27908676956160, 1379187810000000, 69648048665395200, 3619848890071814400, 194428045753727385600, 10817313676715756770560, 624148118196535296000000, 37365232440634029661766400

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=9 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+9)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= a->add(b(j, 9)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);

Mathematica

CoefficientList[Series[1/(1+LambertW[-x]^9)^(1/9),{x,0,20}],x] * Range[0,20]!  (* Vaclav Kotesovec, Sep 01 2014 *)

Formula

E.g.f.: 1/(1+LambertW(-x)^9)^(1/9). - Vaclav Kotesovec, Sep 01 2014

a(n) ~ n^(n-4/9) * 2^(4/9) * sqrt(Pi) / (3^(2/9) * Gamma(1/18)) * (1 - 52 * sqrt(2/n) * Gamma(1/18) / (243 * Gamma(5/9))). - Vaclav Kotesovec, Sep 01 2014