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A051337 Number of strongly connected tournaments on n nodes.

%I A051337 #25 Sep 12 2019 08:14:23
%S A051337 1,1,0,1,1,6,35,353,6008,178133,9355949,884464590,152310149735,
%T A051337 48234782263293,28304491788158056,30964247546702883729,
%U A051337 63468402142317299907481,244785748571033855024746438,1782909084196274276970660380187,24602074618353524534591008760307017
%N A051337 Number of strongly connected tournaments on n nodes.
%C A051337 A tournament is strongly connected (or strong) if there is a directed path between any pair of points.
%D A051337 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 127, Eq. (5.2.4);
%D A051337 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 523.
%H A051337 Andrew Howroyd, <a href="/A051337/b051337.txt">Table of n, a(n) for n = 0..50</a>
%H A051337 John W. Moon, <a href="http://www.gutenberg.org/ebooks/42833">Topics on tournaments</a>, Holt, Rinehard and Winston (1968)
%H A051337 Peter Steinbach, <a href="/A000664/a000664_11.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 11 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
%H A051337 Raphael Yuster, <a href="https://doi.org/10.1137/1.9781611975482.75">Vector clique decompositions</a>, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms (2019), 1221-1238.
%F A051337 G.f.: = 2 - 1/B(x) where B(x) = g.f. for A000568.
%t A051337 m = 20;
%t A051337 permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
%t A051337 edges[v_] := Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[Quotient[v[[i]], 2], {i, 1, Length[v]}];
%t A051337 oddp[v_] := (For[i = 1, i <= Length[v], i++, If[BitAnd[v[[i]], 1] == 0, Return[0]]]; 1);
%t A051337 b[n_] := b[n] = (s = 0; Do[If[oddp[p] == 1, s += permcount[p]*2^edges[p]], {p, IntegerPartitions[n]}]; s/n!);
%t A051337 B[x_] = Sum[b[k] x^k, {k, 0, m}];
%t A051337 A[x_] = 2 - 1/B[x];
%t A051337 A[x] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Sep 12 2019, after _Andrew Howroyd_ in A000568 *)
%Y A051337 Cf. A000568, A054946.
%K A051337 nonn,nice,easy
%O A051337 0,6
%A A051337 _Vladeta Jovovic_
%E A051337 a(0)=1 prepended and a(18)-a(19) from _Andrew Howroyd_, Sep 10 2018