A202013 The number of functions f:{1,2,...,n}->{1,2,...,n} that have an odd number of odd length cycles and no even length cycles.

0, 1, 2, 12, 100, 1120, 15606, 260344, 5056136, 112026240, 2788230250, 77009739136, 2337124786668, 77302709780608, 2767629599791070, 106631592312384000, 4398877912885363216, 193450993635808976896, 9034380526387410161874, 446519425974262943518720, 23284829853408862172112500

Offset: 0

Author

Geoffrey Critzer, Dec 08 2011

Keywords

• nonn

Comments

The number of endofunctions with an odd number of recurrent elements.

It appears that almost all endofunctions have an even number of recurrent elements.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..386

Crossrefs

Cf. A060435, A116956.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, t, add(
      `if`(j::odd, (j-1)!*b(n-j, 1-t)*
       binomial(n-1, j-1), 0), j=1..n))
    end:
a:= n-> add(b(j, 0)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 20 2016

Mathematica

t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Range[0, 20]! CoefficientList[Series[Sinh[Log[((1 + t)/(1 - t))^(1/2)]], {x, 0, 20}], x]
CoefficientList[Series[(((1-LambertW[-x])/(1+LambertW[-x]))^(1/2))/2 - 1/(2*((1-LambertW[-x])/(1+LambertW[-x]))^(1/2)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 24 2013 *)

Formula

E.g.f.: sinh(log(((1-LambertW(-x))/(1+LambertW(-x)))^(1/2))). - corrected by Vaclav Kotesovec, Sep 24 2013

a(n) ~ n! * 2^(3/4)*Gamma(3/4)*exp(n)/(4*Pi*n^(3/4)) * (1+7*Pi/(24*Gamma(3/4)^2*sqrt(n))). - Vaclav Kotesovec, Sep 24 2013