This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382223 #65 Mar 25 2025 12:57:57
%S A382223 1,0,1,0,1,1,0,1,2,1,0,1,5,3,1,0,1,16,12,4,1,0,1,67,66,22,5,1,0,1,374,
%T A382223 513,172,35,6,1,0,1,2825,5769,1969,355,51,7,1,0,1,29212,95706,33856,
%U A382223 5380,636,70,8,1,0,1,417199,2379348,893188,125090,12006,1036,92,9,1
%N A382223 Rectangular array read by antidiagonals: T(n,k) is the number of labeled digraphs on [n] along with a (coloring) function c:[n] -> [k] with the property that for all u,v in [n], u->v implies u<v and c(u)<c(v), n>=0, k>=0.
%H A382223 Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, <a href="https://doi.org/10.1016/j.disc.2020.112041">Counting acyclic and strong digraphs by descents</a>, Discrete Mathematics, Vol. 343, No. 11 (2020), 112041; <a href="https://arxiv.org/abs/1909.01550">arXiv preprint</a>, arXiv:1909.01550 [math.CO], 2019-2020. See Table 2.
%H A382223 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/pubs/pubfiles/18.pdf">Acyclic orientation of graphs,</a> Discrete Math. 5 (1973), 171-178. North Holland Publishing Company.
%F A382223 Sum_{n>=0} T(n,k)/A005329(n) = e(x)^k, where e(x) = Sum_{n>=0}x^n/A005329(n).
%e A382223 1, 1, 1, 1, 1, 1, 1,...
%e A382223 0, 1, 2, 3, 4, 5, 6,...
%e A382223 0, 1, 5, 12, 22, 35, 51,...
%e A382223 0, 1, 16, 66, 172, 355, 636,...
%e A382223 0, 1, 67, 513, 1969, 5380, 12006,...
%e A382223 0, 1, 374, 5769, 33856, 125090, 352476,...
%t A382223 nn = 6; B[n_] := QFactorial[n, 2];e[z_] := Sum[z^n/B[n], {n, 0, nn}]; zetapolys = Drop[Map[Expand[InterpolatingPolynomial[#, x]] &,Transpose[Table[Table[B[n], {n, 0, nn}] CoefficientList[Series[e[z]^k, {z, 0, nn}], z], {k, 1, nn}]]], -1];Table[zetapolys /. x -> i, {i, 0, nn}] // Transpose // Grid
%Y A382223 Cf. A006116 column k=2, A289539 column k=3, A005329, A382363.
%K A382223 nonn,tabl
%O A382223 0,9
%A A382223 _Geoffrey Critzer_, Mar 23 2025