Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #19 May 20 2016 07:51:36
%S 1,1,3,18,157,1800,25551,432376,8494809,190029888,4768313275,
%T 132626098176,4049755214517,134677876657792,4845193429684167,
%U 187490897290080000,7765153170076158001,342721890859339812864,16058392049508837366771,796093438190851834236928
%N Number of functions f:{1,2,...,n}->{1,2,...,n} with odd cycles only.
%H Alois P. Heinz, <a href="/A116956/b116956.txt">Table of n, a(n) for n = 0..386</a>
%F E.g.f.: sqrt((1-LambertW(-x))/(1+LambertW(-x))).
%F Sum_{k=0..n} binomial(n,k)*a(k)*a(n-k) = 2*n^n, n>0. - _Vladeta Jovovic_, Oct 11 2007
%F a(n) ~ n! * 2^(3/4)*Gamma(3/4)*exp(n)/(2*Pi*n^(3/4)). - _Vaclav Kotesovec_, Sep 24 2013
%p b:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,
%p (j-1)!*b(n-j)*binomial(n-1, j-1), 0), j=1..n))
%p end:
%p a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, May 20 2016
%t t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Range[0, 20]! CoefficientList[
%t Series[((1 + t)/(1 - t))^(1/2), {x, 0, 20}], x] (* _Geoffrey Critzer_, Dec 07 2011 *)
%Y Cf. A070896, A060281, A060435, A070896.
%Y Cf. A212599.
%K easy,nonn
%O 0,3
%A _Vladeta Jovovic_, Mar 30 2006