A354607 - OEIS

%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