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A361579 Triangular array read by rows. T(n,k) is the number of labeled digraphs on [n] with exactly k source-like components, n >= 0, 0 <= k <= n.

%I A361579 #14 Mar 26 2023 11:35:25
%S A361579 1,0,1,0,3,1,0,51,12,1,0,3614,447,34,1,0,991930,53675,2885,85,1,0,
%T A361579 1051469032,21514470,741455,16665,201,1,0,4366988803688,30405612790,
%U A361579 642187105,9816380,90678,462,1,0,71895397383029040,160152273169644,2024633081100,19625842425,122330544,474138,1044,1
%N A361579 Triangular array read by rows. T(n,k) is the number of labeled digraphs on [n] with exactly k source-like components, n >= 0, 0 <= k <= n.
%C A361579 Here, a source-like component of a digraph D is a strongly connected component of D that corresponds to a node of in-degree 0 in the condensation of D.
%H A361579 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 A361579 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 A361579 Wikipedia, <a href="https://en.wikipedia.org/wiki/Strongly_connected_component">Strongly connected component</a>
%e A361579 Triangle begins:
%e A361579   1;
%e A361579   0,      1;
%e A361579   0,      3,     1;
%e A361579   0,     51,    12,    1;
%e A361579   0,   3614,   447,   34,  1;
%e A361579   0, 991930, 53675, 2885, 85, 1;
%e A361579   ...
%t A361579 nn = 6; B[n_] := n! 2^Binomial[n, 2]; strong =Select[Import["https://oeis.org/A003030/b003030.txt", "Table"], Length@# == 2 &][[All, 2]]; s[x_] := Total[strong Table[x^i/i!, {i, 1, 58}]];
%t A361579 ggfz[egfx_] := Normal[Series[egfx, {x, 0, nn}]] /.Table[x^i -> z^i/2^Binomial[i, 2], {i, 0, nn}];Table[B[n], {n, 0, nn}] CoefficientList[Series[ggfz[Exp[(u - 1) s[x]]]/ggfz[Exp[- s[x]]], {z, 0, nn}], {z u}] // Grid
%Y A361579 Cf. A003028 (column k=1), A053763 (row sums).
%K A361579 nonn,tabl
%O A361579 0,5
%A A361579 _Geoffrey Critzer_, Mar 16 2023