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A362013 Triangular array read by rows. T(n,k) is the number of labeled directed graphs on [n] with exactly k strongly connected components of size 1 with outdegree zero, n>=0, 0<=k<=n.

%I A362013 #12 Apr 05 2023 06:37:41
%S A362013 1,0,1,1,2,1,27,27,9,1,2401,1372,294,28,1,759375,253125,33750,2250,75,
%T A362013 1,887503681,171774906,13852815,595820,14415,186,1,3938980639167,
%U A362013 437664515463,20841167403,551353635,8751645,83349,441,1,67675234241018881,4263006881324024,117484441611292,1850148686792,18210124870,114709448,451612,1016,1
%N A362013 Triangular array read by rows.  T(n,k) is the number of labeled directed graphs on [n] with exactly k strongly connected components of size 1 with outdegree zero, n>=0, 0<=k<=n.
%H A362013 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.
%e A362013 Triangle T(n,k) begins:
%e A362013        1;
%e A362013        0,      1;
%e A362013        1,      2,     1;
%e A362013       27,     27,     9,    1;
%e A362013     2401,   1372,   294,   28,  1;
%e A362013   759375, 253125, 33750, 2250, 75, 1;
%e A362013   ...
%t A362013 nn = 6; B[n_] := n! 2^Binomial[n, 2] ; strong =Select[Import["https://oeis.org/A003030/b003030.txt", "Table"], Length@# == 2 &][[All, 2]]; s[z_] := Total[strong Table[z^i/i!, {i, 1, 58}]];
%t A362013 ggf[egf_] := Normal[Series[egf, {z, 0, nn}]] /.Table[z^i -> z^i/2^Binomial[i, 2], {i, 0, nn}]; Table[ Take[(Table[B[n], {n, 0, nn}] CoefficientList[   Series[ggf[Exp[(u - 1) z]]/ggf[Exp[-s[z]]], {z, 0, nn}], {z, u}])[[i]], i], {i, 1, nn + 1}]
%Y A362013 Cf. A086206 (column k=0), A053763 (row sums), A361592, A350792 (a subclass of the digraphs for the case k=1 of this sequence), A003028.
%K A362013 nonn,tabl
%O A362013 0,5
%A A362013 _Geoffrey Critzer_, Apr 03 2023