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A219694 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n} -> {1,2,...,n} that have exactly k nonrecurrent elements; n>=1, 0<=k<=n-1.

%I A219694 #15 May 22 2016 13:11:31
%S A219694 1,2,2,6,12,9,24,72,96,64,120,480,900,1000,625,720,3600,8640,12960,
%T A219694 12960,7776,5040,30240,88200,164640,216090,201684,117649,40320,282240,
%U A219694 967680,2150400,3440640,4128768,3670016,2097152,362880,2903040,11430720,29393280,55112400,79361856,89282088,76527504,43046721
%N A219694 Triangular array read by rows:  T(n,k) is the number of functions f:{1,2,...,n} -> {1,2,...,n} that have exactly k nonrecurrent elements; n>=1, 0<=k<=n-1.
%C A219694 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.  An element that is not recurrent is a nonrecurrent element.
%H A219694 Alois P. Heinz, <a href="/A219694/b219694.txt">Rows n = 1..141, flattened</a>
%F A219694 E.g.f.: 1/(1-x*exp(A(x,y))), where A(x,y) = Sum_{n>=1} n^(n-1)*(y*x)^n/n!.
%e A219694 T(2,1) = 2 because we have 1->1 2->1; and 1->2 2->2.
%e A219694 :    1;
%e A219694 :    2,     2;
%e A219694 :    6,    12,     9;
%e A219694 :   24,    72,    96,     64;
%e A219694 :  120,   480,   900,   1000,    625;
%e A219694 :  720,  3600,  8640,  12960,  12960,   7776;
%e A219694 : 5040, 30240, 88200, 164640, 216090, 201684, 117649;
%p A219694 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A219694       (j-1)!*b(n-j)*binomial(n-1, j-1), j=1..n))
%p A219694     end:
%p A219694 T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(add(
%p A219694     b(j)*(x*n)^(n-j)*binomial(n-1, j-1), j=0..n)):
%p A219694 seq(T(n), n=1..10);  # _Alois P. Heinz_, May 22 2016
%t A219694 nn=8;f[list_]:=Select[list,#>0&];t=Sum[n^(n-1)x^n y^n/n!,{n,1,nn}];Drop[Map[f,Range[0,nn]!CoefficientList[Series[1/(1-x Exp[t]),{x,0,nn}],{x,y}]],1]//Grid
%Y A219694 Cf. A216971.
%K A219694 nonn,tabl
%O A219694 1,2
%A A219694 _Geoffrey Critzer_, Nov 25 2012