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A204042 The number of functions f:{1,2,...,n}->{1,2,...,n} (endofunctions) such that all of the fixed points in f are isolated.

%I A204042 #25 Dec 31 2021 14:01:47
%S A204042 1,1,2,12,120,1520,23160,413952,8505280,197631072,5125527360,
%T A204042 146787894440,4601174623584,156693888150384,5761055539858528,
%U A204042 227438694372072120,9596077520725211520,430920897407809702208,20520683482765477749120,1032920864149903149579336,54797532208320308334631840
%N A204042 The number of functions f:{1,2,...,n}->{1,2,...,n} (endofunctions) such that all of the fixed points in f are isolated.
%C A204042 Note this sequence counts the functions enumerated by A065440 for which the statement is vacuously true.
%C A204042 a(n) is also the number of partial endofunctions on {1,2,...,n} without fixed points.
%H A204042 Alois P. Heinz, <a href="/A204042/b204042.txt">Table of n, a(n) for n = 0..386</a>
%F A204042 E.g.f.: exp(x)*A(x) where A(x) is the e.g.f. for A065440.
%F A204042 a(n) ~ exp(exp(-1)-1)*n^n. - _Vaclav Kotesovec_, Sep 24 2013
%F A204042 a(n) = Sum_{j=0..n} (j-1)^j * binomial(n,j). - _Alois P. Heinz_, Dec 16 2021
%e A204042 a(2)=2 because there are two functions f:{1,2}->{1,2} in which all the fixed points are isolated: 1->1,2->2  and 1->2,2->1 (which has no fixed points).
%p A204042 a:= n-> add((j-1)^j*binomial(n, j), j=0..n):
%p A204042 seq(a(n), n=0..20);  # _Alois P. Heinz_, Dec 16 2021
%t A204042 t = Sum[n^(n-1) x^n/n!, {n,1,20}]; Range[0,20]! CoefficientList[Series[Exp[x] Exp[Log[1/(1-t)]-t], {x,0,20}], x]
%Y A204042 Cf. A065440, A086331, A350134.
%Y A204042 Row sums of A349454.
%K A204042 nonn
%O A204042 0,3
%A A204042 _Geoffrey Critzer_, Jan 09 2012