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A203092 Triangular array read by rows. T(n,k) is the number of partial functions on {1,2,...,n} that are endofunctions with no cycles of length > 1 that have exactly k components.

%I A203092 #13 Jan 20 2014 22:13:31
%S A203092 1,1,1,1,4,1,1,18,9,1,1,116,78,16,1,1,1060,810,220,25,1,1,12702,10335,
%T A203092 3260,495,36,1,1,187810,158613,54740,9835,966,49,1,1,3296120,2854908,
%U A203092 1046024,209510,24696,1708,64,1
%N A203092 Triangular array read by rows.  T(n,k) is the number of partial functions on {1,2,...,n} that are endofunctions with no cycles of length > 1 that have exactly k components.
%C A203092 Row sums = A088957.
%C A203092 T(n,0)= 1,  the empty function.
%C A203092 T(n,n)= 1,  the identity function.
%C A203092 T(n,n-1)= n^2  (apparently).
%F A203092 E.g.f.: exp(x)*exp(y T(x)) where T(x) is the e.g.f. for A000169.
%e A203092 T(2,1)= 4 because there are 4 such partial functions on {1,2}: 1->1, 2->2, 1->1 2->1, 1->2 2->2,
%e A203092 1
%e A203092 1     1
%e A203092 1     4     1
%e A203092 1     18    9     1
%e A203092 1     116   78    16    1
%e A203092 1     1060  810   220   25    1
%e A203092 1     12702 10335 3260  495   36    1
%t A203092 nn = 8; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
%t A203092 f[list_] := Select[list, # > 0 &];
%t A203092 Map[f, Range[0, nn]! CoefficientList[ Series[Exp[x] Exp[y t], {x, 0, nn}], {x, y}]] // Flatten
%Y A203092 Cf. A088956, A144289
%K A203092 nonn,tabl
%O A203092 0,5
%A A203092 _Geoffrey Critzer_, Dec 29 2011