%I M3404 #26 Feb 20 2023 22:08:38

%S 1,1,4,10,136,720,44224,703760,179228736,9168331776,9383939974144,

%T 1601371799340544,6558936236286040064,3837878966366932639744,

%U 62879572771326489528942592,128777257564337108286016980992

%N Number of self-complementary digraphs with n nodes.

%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, pp. 140, 243.

%D R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Andrew Howroyd, <a href="/A003086/b003086.txt">Table of n, a(n) for n = 1..80</a>

%t Table[GraphPolynomial[n,x,Directed]/.x -> -1,{n,1,20}] (* _Geoffrey Critzer_, Oct 21 2012 *)

%t 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 edges[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 a[n_] := (s = 0; Do[s += permcount[2 p]*2^edges[p]*If[OddQ[n], n *4^Length[p], 1], {p, IntegerPartitions[n/2 // Floor]}]; s/n!);

%t Array[a, 16] (* _Jean-François Alcover_, Aug 26 2019, after _Andrew Howroyd_ *)

%o (PARI)

%o permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}

%o edges(v) = {4*sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, 2*v[i]-1)}

%o a(n) = {my(s=0); forpart(p=n\2, s+=permcount(2*Vec(p))*2^edges(p)*if(n%2, n*4^#p, 1)); s/n!} \\ _Andrew Howroyd_, Sep 16 2018

%Y Cf. A000171, A002785.

%K nonn,nice,easy

%O 1,3

%A _N. J. A. Sloane_

%E More terms from Ronald C. Read and _Vladeta Jovovic_.