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A116956 Number of functions f:{1,2,...,n}->{1,2,...,n} with odd cycles only.

%I A116956 #19 May 20 2016 07:51:36
%S A116956 1,1,3,18,157,1800,25551,432376,8494809,190029888,4768313275,
%T A116956 132626098176,4049755214517,134677876657792,4845193429684167,
%U A116956 187490897290080000,7765153170076158001,342721890859339812864,16058392049508837366771,796093438190851834236928
%N A116956 Number of functions f:{1,2,...,n}->{1,2,...,n} with odd cycles only.
%H A116956 Alois P. Heinz, <a href="/A116956/b116956.txt">Table of n, a(n) for n = 0..386</a>
%F A116956 E.g.f.: sqrt((1-LambertW(-x))/(1+LambertW(-x))).
%F A116956 Sum_{k=0..n} binomial(n,k)*a(k)*a(n-k) = 2*n^n, n>0. - _Vladeta Jovovic_, Oct 11 2007
%F A116956 a(n) ~ n! * 2^(3/4)*Gamma(3/4)*exp(n)/(2*Pi*n^(3/4)). - _Vaclav Kotesovec_, Sep 24 2013
%p A116956 b:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,
%p A116956        (j-1)!*b(n-j)*binomial(n-1, j-1), 0), j=1..n))
%p A116956     end:
%p A116956 a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A116956 seq(a(n), n=0..20);  # _Alois P. Heinz_, May 20 2016
%t A116956 t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Range[0, 20]! CoefficientList[
%t A116956 Series[((1 + t)/(1 - t))^(1/2), {x, 0, 20}], x]  (* _Geoffrey Critzer_, Dec 07 2011 *)
%Y A116956 Cf. A070896, A060281, A060435, A070896.
%Y A116956 Cf. A212599.
%K A116956 easy,nonn
%O A116956 0,3
%A A116956 _Vladeta Jovovic_, Mar 30 2006