%I #19 Apr 17 2024 20:47:03
%S 1,1,2,6,18,78,426,2286,15402,122190,951546,8724078,90768378,
%T 928340190,10779805722,138779942046,1759271695338,24739709631678,
%U 379578822373866,5743346972756526,94864142045862282,1689637343582548590,29717468115957434586,563879701735681033998
%N Number of acyclic orientations of the Turán graph T(n,3).
%C An acyclic orientation is an assignment of a direction to each edge such that no cycle in the graph is consistently oriented. Stanley showed that the number of acyclic orientations of a graph G is equal to the absolute value of the chromatic polynomial X_G(q) evaluated at q=-1.
%H Alois P. Heinz, <a href="/A266858/b266858.txt">Table of n, a(n) for n = 0..465</a>
%H P. J. Cameron, C. A. Glass, R. U. Schumacher, <a href="https://arxiv.org/abs/1412.3685">Acyclic orientations and poly-Bernoulli numbers</a>, arXiv:1412.3685 [math.CO], 2014-2018.
%H Richard P. Stanley, <a href="http://dx.doi.org/10.1016/0012-365X(73)90108-8">Acyclic Orientations of Graphs</a>, Discrete Mathematics, 5 (1973), pages 171-178, doi:10.1016/0012-365X(73)90108-8
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tur%C3%A1n_graph">Turán graph</a>
%F a(n) ~ n! / (2*(1 - log(3/2)) * 3^n * (log(3/2))^(n+1)). - _Vaclav Kotesovec_, Feb 18 2017
%t A[n_, k_] := A[n, k] = Module[{b, l, q}, q = -1; l = Join[Array[Floor[ n/k]&, k - Mod[n, k]], Array[Ceiling[n/k]&, Mod[n, k]]]; b[nn_, j_] := b[nn, j] = If[j==1, (q-nn)^l[[1]] Product[q-i, {i, 0, nn-1}], Sum[b[nn + m, j-1] StirlingS2[l[[j]], m], {m, 0, l[[j]]}]]; Abs[b[0, k]]];
%t a[n_] := A[n, 3];
%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Aug 20 2018, after _Alois P. Heinz_ *)
%Y Column k=3 of A267383.
%Y Cf. A182553, A212220, A266695.
%Y Trisection gives A370961.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jan 04 2016