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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. 1
0, 1, 2, 12, 100, 1120, 15606, 260344, 5056136, 112026240, 2788230250, 77009739136, 2337124786668, 77302709780608, 2767629599791070, 106631592312384000, 4398877912885363216, 193450993635808976896, 9034380526387410161874, 446519425974262943518720, 23284829853408862172112500 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A168365 A055865 A085389 * A151505 A096347 A137483
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Dec 08 2011
STATUS
approved

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Last modified April 12 16:45 EDT 2024. Contains 371635 sequences. (Running on oeis4.)