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%I #17 May 20 2016 05:56:36
%S 1,0,2,6,60,560,7350,111552,2009672,41378976,963527850,25009038560,
%T 716437784172,22453784964624,764345507271710,28085186967504240,
%U 1107971902218683280,46710909213378892352,2095883952368863510098,99724281567446320231104,5015524096516005263567540
%N The number of functions f:{1,2,...,n}->{1,2,...,n}, endofunctions, such that exactly one nonrecurrent element is mapped into each recurrent element.
%C x in {1,2,...,n} is a recurrent element if there is some k such that f^k(x) = x where f^k(x) denotes iterated functional composition. In other words, a recurrent element is in a cycle of the functional digraph.
%H Alois P. Heinz, <a href="/A215720/b215720.txt">Table of n, a(n) for n = 0..150</a>
%F E.g.f.: 1/(1 - x*T(x)) where T(x) is the e.g.f. for A000169.
%F a(n) = n! * Sum_{i=0..floor(n/2)} i*(n-i)^(n-2*i-1)/(n-2*i)! for n>0, a(0) = 1. - _Alois P. Heinz_, Aug 22 2012
%F a(n) ~ exp(1)/(exp(1)-1)^2 * n^(n-1). - _Vaclav Kotesovec_, Sep 30 2013
%e a(2) = 2 because we have: (1->1,2->1), (1->2,2->2).
%p a:= n-> `if`(n=0, 1, n! *add(i*(n-i)^(n-2*i-1)/(n-2*i)!, i=0..n/2)):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Aug 22 2012
%t nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
%t Range[0, nn]! CoefficientList[Series[1/(1 - x t) , {x, 0, nn}], x]
%Y Cf. A055541, A195203, A098875.
%K nonn
%O 0,3
%A _Geoffrey Critzer_, Aug 22 2012
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