cp's OEIS Frontend

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.

%I A202013 #20 May 20 2016 07:57:36
%S A202013 0,1,2,12,100,1120,15606,260344,5056136,112026240,2788230250,
%T A202013 77009739136,2337124786668,77302709780608,2767629599791070,
%U A202013 106631592312384000,4398877912885363216,193450993635808976896,9034380526387410161874,446519425974262943518720,23284829853408862172112500
%N 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.
%C A202013 The number of endofunctions with an odd number of recurrent elements.
%C A202013 It appears that almost all endofunctions have an even number of recurrent elements.
%H A202013 Alois P. Heinz, <a href="/A202013/b202013.txt">Table of n, a(n) for n = 0..386</a>
%F A202013 E.g.f.: sinh(log(((1-LambertW(-x))/(1+LambertW(-x)))^(1/2))). - corrected by _Vaclav Kotesovec_, Sep 24 2013
%F A202013 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
%p A202013 b:= proc(n, t) option remember; `if`(n=0, t, add(
%p A202013       `if`(j::odd, (j-1)!*b(n-j, 1-t)*
%p A202013        binomial(n-1, j-1), 0), j=1..n))
%p A202013     end:
%p A202013 a:= n-> add(b(j, 0)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A202013 seq(a(n), n=0..20);  # _Alois P. Heinz_, May 20 2016
%t A202013 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]
%t A202013 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 *)
%Y A202013 Cf. A060435, A116956.
%K A202013 nonn
%O A202013 0,3
%A A202013 _Geoffrey Critzer_, Dec 08 2011