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%I #23 Mar 16 2023 04:50:35
%S 1,0,1,0,1,3,0,2,21,25,0,6,213,774,543,0,24,3470,30275,59830,29281,0,
%T 120,95982,1847265,7757355,10110735,3781503,0,720,4578588,190855000,
%U 1522899105,3944546095,3767987307,1138779265
%N Triangular array read by rows. T(n,k) is the number of labeled digraphs on [n] having exactly k strongly connected components all of which are simple cycles, n >= 0, 0 <= k <= n.
%C Here, a strongly connected component containing exactly 1 vertex is considered a cycle.
%H E. de Panafieu and S. Dovgal, <a href="https://arxiv.org/abs/1903.09454">Symbolic method and directed graph enumeration</a>, arXiv:1903.09454 [math.CO], 2019.
%H R. W. Robinson, <a href="http://cobweb.cs.uga.edu/~rwr/publications/components.pdf">Counting digraphs with restrictions on the strong components</a>, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SimpleDirectedGraph.html">Simple Directed Graph</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Strongly_connected_component">Strongly connected component</a>
%e 1;
%e 0, 1;
%e 0, 1, 3;
%e 0, 2, 21, 25;
%e 0, 6, 213, 774, 543;
%e 0, 24,3470, 30275, 59830, 29281;
%e ...
%t nn = 7;
%t a[x_] := Log[1/(1 - x)];
%t begfa =Total[CoefficientList[ Series[1/(Total[ CoefficientList[Series[ Exp[-u *a[x]], {x, 0, nn}], x]* Table[z^n/(2^Binomial[n, 2]), {n, 0, nn}]]), {z, 0, nn}], z]*Table[z^n 2^Binomial[n, 2], {n, 0, nn}]];
%t Table[Take[(Range[0, nn]! CoefficientList[begfa, {z, u}])[[i]],i], {i, 1, nn + 1}] // Grid
%Y Cf. A011266 (row sums), A003024 (main diagonal), A000142 (column k=1).
%K nonn,tabl
%O 0,6
%A _Geoffrey Critzer_, Mar 14 2023
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