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%I #17 Jul 12 2022 17:42:34
%S 1,0,1,0,0,2,0,2,0,6,0,24,16,0,24,0,544,240,120,0,120,0,22320,6608,
%T 2160,960,0,720,0,1677488,315840,70224,20160,8400,0,5040,0,236522496,
%U 27001984,3830400,758016,201600,80640,0,40320,0,64026088576,4268194560,366729600,46448640,8628480,2177280,846720,0,362880
%N Triangular array read by rows: T(n,k) is the number of labeled tournaments on [n] that have exactly k irreducible (strongly connected) components, n >= 0, 0 <= k <= n.
%H N. J. A. Sloane, <a href="/A000568/a000568_1.pdf">Illustration of first 5 terms</a>
%H Peter Steinbach, <a href="/A000664/a000664_11.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 11 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
%F E.g.f.: 1/(1-y*(1-1/A(x))) where A(x) is the e.g.f. for A006125.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 0, 2;
%e 0, 2, 0, 6;
%e 0, 24, 16, 0, 24;
%e 0, 544, 240, 120, 0, 120;
%e 0, 22320, 6608, 2160, 960, 0, 720;
%e ...
%t nn = 10; G[x_] := Sum[2^Binomial[n, 2] x^n/n!, {n, 0, nn}]; Table[
%t Take[(Range[0, nn]! CoefficientList[Series[1/(1 - y (1 - 1/ G[x])), {x, 0, nn}], {x, y}])[[i]], i], {i, 1, nn}]
%Y Cf. A006125 (row sums), A054946 (column k=1), A000142 (main diagonal).
%K nonn,tabl
%O 0,6
%A _Geoffrey Critzer_, Jul 08 2022
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