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A209319 Number of functions f:{1,2,...,n}->{1,2,...,n} whose cycle lengths are <= 2.

%I A209319 #20 Aug 28 2014 16:58:32
%S A209319 1,1,4,25,218,2451,33832,554527,10535100,227790505,5525843696,
%T A209319 148673435769,4394818486456,141611317636075,4940870266568160,
%U A209319 185595910032346111,7468517348971708688,320562141349559055633,14619577651630443611200,706025600924216704982425
%N A209319 Number of functions f:{1,2,...,n}->{1,2,...,n} whose cycle lengths are <= 2.
%H A209319 Alois P. Heinz, <a href="/A209319/b209319.txt">Table of n, a(n) for n = 0..350</a>
%F A209319 E.g.f.: exp(T(x) + T(x)^2/2) = A(T(x)) where A(x) is the e.g.f. for A000085 and T(x) is the e.g.f. for A000169.
%F A209319 a(n) ~ 2*exp(3/2)*n^(n-1). - _Vaclav Kotesovec_, Sep 30 2013
%e A209319 a(3) = 25 because there are 27 functions from {1,2,3} into itself but 2 of these have cycle length of 3: 2,3,1, and 3,1,2.
%p A209319 T:= -LambertW(-x):
%p A209319 egf:= exp(T + T^2/2):
%p A209319 a:= n-> n!*coeff(series(egf, x, n+1), x, n):
%p A209319 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jan 19 2013
%t A209319 nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[Exp[Sum[t^i/i,{i,1,2}]],{x,0,nn}],x]
%Y A209319 Column k=2 of A246522.
%K A209319 nonn
%O A209319 0,3
%A A209319 _Geoffrey Critzer_, Jan 19 2013