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.

1, 2, 10, 114, 4872, 770832, 441038832, 896679948304, 6513978501814144, 170630215981070456064, 16261454692532635025585792, 5683372715412701087902846672384, 7334542846356464937798016155801130496, 35157828307617499760694672217473135511928832

Offset: 1

Author

N. J. A. Sloane, May 24 2000

Keywords

• nonn
• easy

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..50
• V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Crossrefs

Cf. A000273, A003086, A007869, A054960, A055969.

Programs

Mathematica

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];
edges[v_] := Sum[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v - 1];
b[n_] := (s = 0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
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]}];
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!);
a[n_] := (b[n] + c[n])/2;
Array[a, 14] (* Jean-François Alcover, Aug 26 2019, using Andrew Howroyd's code for b=A000273 and c=A003086 *)

Formula

Average of A000273 and A003086.

Extensions

More terms from Vladeta Jovovic, Jul 19 2000

Terms a(14) and beyond from Andrew Howroyd, Sep 17 2018