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A054928 Number of complementary pairs of directed graphs on n nodes. Also number of unlabeled digraphs with n nodes and an even number of arcs.

%I A054928 #20 Aug 26 2019 08:05:38
%S A054928 1,2,10,114,4872,770832,441038832,896679948304,6513978501814144,
%T A054928 170630215981070456064,16261454692532635025585792,
%U A054928 5683372715412701087902846672384,7334542846356464937798016155801130496,35157828307617499760694672217473135511928832
%N A054928 Number of complementary pairs of directed graphs on n nodes. Also number of unlabeled digraphs with n nodes and an even number of arcs.
%H A054928 Andrew Howroyd, <a href="/A054928/b054928.txt">Table of n, a(n) for n = 1..50</a>
%H A054928 V. A. Liskovets, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LISK/Derseq.html">Some easily derivable sequences</a>, J. Integer Sequences, 3 (2000), #00.2.2.
%F A054928 Average of A000273 and A003086.
%t A054928 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 A054928 edges[v_] := Sum[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v - 1];
%t A054928 b[n_] := (s = 0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
%t A054928 edges4[v_] := 4 Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[2 v[[i]] - 1, {i, 1, Length[v]}];
%t A054928 c[n_] := (s = 0; Do[s += permcount[2 p]*2^edges4[p]*If[OddQ[n], n *4^Length[p], 1], {p, IntegerPartitions[n/2 // Floor]}]; s/n!);
%t A054928 a[n_] := (b[n] + c[n])/2;
%t A054928 Array[a, 14] (* _Jean-François Alcover_, Aug 26 2019, using _Andrew Howroyd_'s code for b=A000273 and c=A003086 *)
%Y A054928 Cf. A000273, A003086, A007869, A054960, A055969.
%K A054928 nonn,easy
%O A054928 1,2
%A A054928 _N. J. A. Sloane_, May 24 2000
%E A054928 More terms from _Vladeta Jovovic_, Jul 19 2000
%E A054928 Terms a(14) and beyond from _Andrew Howroyd_, Sep 17 2018