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

0,3

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

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

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 *)

Crossrefs

Cf. A060435, A116956.

Sequence in context: A168365 A055865 A085389 * A151505 A096347 A137483

Adjacent sequences: A202010 A202011 A202012 * A202014 A202015 A202016

Keyword

nonn

Author

Geoffrey Critzer, Dec 08 2011

Status

approved