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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.
1
1, 0, 1, 1, 2, 1, 27, 27, 9, 1, 2401, 1372, 294, 28, 1, 759375, 253125, 33750, 2250, 75, 1, 887503681, 171774906, 13852815, 595820, 14415, 186, 1, 3938980639167, 437664515463, 20841167403, 551353635, 8751645, 83349, 441, 1, 67675234241018881, 4263006881324024, 117484441611292, 1850148686792, 18210124870, 114709448, 451612, 1016, 1
OFFSET
0,5
LINKS
E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.
EXAMPLE
Triangle T(n,k) begins:
1;
0, 1;
1, 2, 1;
27, 27, 9, 1;
2401, 1372, 294, 28, 1;
759375, 253125, 33750, 2250, 75, 1;
...
MATHEMATICA
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}]];
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}]
CROSSREFS
Cf. A086206 (column k=0), A053763 (row sums), A361592, A350792 (a subclass of the digraphs for the case k=1 of this sequence), A003028.
Sequence in context: A235031 A138955 A089963 * A336911 A322230 A353910
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Apr 03 2023
STATUS
approved