A005639 - OEIS

%I M1518 #32 Aug 17 2019 09:56:20

%S 1,2,5,18,102,848,12452,265759,10454008,598047612,63620448978,

%T 9974635937844,2905660724913768,1268590412128132389,

%U 1023130650177394611897,1258149993547327488275562,2834863110716120144290954314,9900859865505110360978721901778

%N Number of self-converse oriented graphs with n nodes.

%D R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

%D R. W. Robinson, Asymptotic number of self-converse oriented graphs, pp. 255-266 of Combinatorial Mathematics (Canberra, 1977), Lect. Notes Math. 686, 1978.

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

%H Andrew Howroyd, <a href="/A005639/b005639.txt">Table of n, a(n) for n = 1..50</a> (terms 1..27 from R. W. Robinson)

%H R. W. Robinson, <a href="/A005639/a005639.pdf">Asymptotic number of self-converse oriented graphs</a>, pp. 255-266 of Combinatorial Mathematics (Canberra, 1977), Lect. Notes Math. 686, 1978. (Annotated scanned copy)

%H Sridharan, M. R., <a href="/A002785/a002785.pdf"> Self-complementary and self-converse oriented graphs </a>, Nederl. Akad. Wetensch. Proc. Ser. A 73=Indag. Math. 32 1970 441-447. [Annotated scanned copy] See page 446.

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

%t a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p], {p, IntegerPartitions[n]}]; s/n!];

%t Array[a, 18] (* _Jean-François Alcover_, Aug 17 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, if(v[i]*v[j]%2==0, gcd(v[i],v[j])))) + sum(i=1, #v, if(v[i]%2==0, (v[i]-2)\4*2+1))}

%o a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)); s/n!} \\ _Andrew Howroyd_, Sep 16 2018

%Y Cf. A002785.

%K nonn,easy,nice

%O 1,2

%A _N. J. A. Sloane_