A246611 Number of endofunctions on [n] whose cycle lengths are multiples of 4.

1, 0, 0, 0, 6, 120, 2160, 41160, 866460, 20294064, 526680000, 15036999120, 468848156040, 15859299473160, 578619457031616, 22654279249875000, 947570269816868880, 42174922731482980320, 1990416896317283627520, 99290011292792071612704, 5220362654145754082460000

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=4 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+4)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= a->add(b(j, 4)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);

Mathematica

CoefficientList[Series[1/(1-LambertW[-x]^4)^(1/4),{x,0,20}],x] * Range[0,20]!  (* Vaclav Kotesovec, Sep 01 2014 *)

Formula

E.g.f.: 1/(1-LambertW(-x)^4)^(1/4). - Vaclav Kotesovec, Sep 01 2014

a(n) ~ n^(n-3/8) * (sqrt(Pi) / (2^(1/8) * Gamma(1/8))) * (1 - 11 * sqrt(2/n) * Gamma(1/8) / (64 * Gamma(5/8))). - Vaclav Kotesovec, Sep 01 2014