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A202013
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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.
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1
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0, 1, 2, 12, 100, 1120, 15606, 260344, 5056136, 112026240, 2788230250, 77009739136, 2337124786668, 77302709780608, 2767629599791070, 106631592312384000, 4398877912885363216, 193450993635808976896, 9034380526387410161874, 446519425974262943518720, 23284829853408862172112500
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OFFSET
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0,3
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COMMENTS
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The number of endofunctions with an odd number of recurrent elements.
It appears that almost all endofunctions have an even number of recurrent elements.
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LINKS
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FORMULA
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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
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MAPLE
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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):
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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