%I M2875 N1156 #38 Dec 31 2020 11:02:30

%S 1,3,10,70,708,15224,544152,39576432,5074417616,1296033011648,

%T 604178966756320,556052774253161600,954895322019762585664,

%U 3224152068625567826724224,20610090531322819956330186112

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

%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 155, Table 6.6.1 (but the last entry is wrong).

%D R. W. Robinson, personal communication.

%D R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1980.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H Andrew Howroyd, <a href="/A002499/b002499.txt">Table of n, a(n) for n = 1..50</a> (terms 1..28 from R. W. Robinson)

%H Alastair Farrugia, <a href="http://www.alastairfarrugia.net/sc-graph/sc-graph-survey.pdf">Self-complementary graphs and generalizations: a comprehensive reference</a>, M.Sc. Thesis, University of Malta, August 1999.

%H F. Harary and E. M. Palmer, <a href="http://dx.doi.org/10.1112/S0025579300003910">Enumeration of self-converse digraphs</a>, Mathematika, 13 (1966), 151-157.

%F Asymptotics (R. W. Robinson): a(n) ~ 2^((n^2 - 1)/2) * exp(sqrt(n/2) - n/2 - 1/8) * n^(n/2) / n!, (Farrugia, formula 7.28, p. 199). - _Vaclav Kotesovec_, Dec 31 2020

%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_] := Sum[Sum[GCD[v[[i]], v[[j]]]*If[Mod[v[[i]] v[[j]], 2]==0, 2, 1], {j, 1, i-1}], {i, 2, Length[v]}]+Sum[Quotient[v[[i]], 2] + If[Mod[v[[i]], 2]==0, Quotient[v[[i]]-2, 4]*2+1, 0], {i, 1, Length[v]}];

%t a[n_] := Module[{s=0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];

%t Array[a, 15] (* _Jean-François Alcover_, Aug 16 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) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j])*if(v[i]*v[j]%2==0, 2, 1))) + sum(i=1, #v, v[i]\2 + if(v[i]%2==0, (v[i]-2)\4*2+1))}

%o a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ _Andrew Howroyd_, Sep 18 2018

%Y Cf. A002500.

%K nonn,nice

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Apr 17 2000