%I #31 Jun 13 2022 12:35:22
%S 1,1,1,1,1,1,1,1,3,1,1,1,5,25,1,1,1,7,109,543,1,1,1,9,289,9449,29281,
%T 1,1,1,11,601,63487,3068281,3781503,1,1,1,13,1081,267249,69711361,
%U 3586048685,1138779265,1
%N Square array read by descending antidiagonals. T(n,k) is the number of acyclic k-multidigraphs on n labeled vertices, n>=0,k>=0.
%C Here, a k-multidigraph is a directed graph where up to k arcs (directed edges) are allowed to join vertex pairs. The arcs have no identity, i.e., they are indistinguishable except for the ordered pair of distinct vertices that they join.
%H Seiichi Manyama, <a href="/A339768/b339768.txt">Antidiagonals n = 0..50, flattened</a>
%H R. P. Stanley, <a href="https://doi.org/10.1016/j.disc.2006.03.010">Acyclic orientation of graphs</a>, Discrete Math. 5 (1973), 171-178.
%F Let E(x) = Sum_{n>=0} x^n/(n!*(k+1)^binomial(n,2)). Then 1/E(-x) = Sum_{n>=0} T(n,k)x^n/(n!*(k+1)^binomial(n,2)).
%F T(0,k) = 1 and T(n,k) = Sum_{j=1..n} (-1)^(j+1) * (k+1)^(j*(n-j)) * binomial(n,j) * T(n-j,k) for n > 0. - _Seiichi Manyama_, Jun 13 2022
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 3, 5, 7, 9, 11, ...
%e 1, 25, 109, 289, 601, 1081, ...
%e 1, 543, 9449, 63487, 267249, 849311, ...
%e 1, 29281, 3068281, 69711361, 742650001, 5004309601, ...
%t nn = 5; Table[g[n_] := q^Binomial[n, 2] n!; e[z_] := Sum[z^k/g[k], {k, 0, nn}];
%t Table[g[n], {n, 0, nn}] CoefficientList[Series[1/e[-z], {z, 0, nn}], z], {q, 1, nn + 1}] //Transpose // Grid
%Y Cf. A003024 (column k=1), A188457 (column k=2), A137435 (column k=3).
%Y Main diagonal gives A354962.
%K nonn,tabl
%O 0,9
%A _Geoffrey Critzer_, Feb 21 2021