A052112 - OEIS

%I #15 Sep 12 2019 09:10:48

%S 1,2,14,159,7629,599456,226066304,139178815861,410179495378288,

%T 2055126126323159298,48234291396964332998082,

%U 2016523952125103590736221923,382812826011951187177138562992638,135681830960694827549160289095792266106

%N Number of self-complementary directed 2-multigraphs on n nodes.

%C A 2-multigraph is similar to an ordinary graph except there are 0, 1 or 2 edges between any two nodes (self-loops are not allowed).

%H Andrew Howroyd, <a href="/A052112/b052112.txt">Table of n, a(n) for n = 1..50</a>

%t permcount[v_List] := 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_List] := 2 Sum[Sum[If[EvenQ[v[[i]] v[[j]]], GCD[v[[i]], v[[j]]], 0], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[If[EvenQ[v[[i]]], v[[i]] - 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, 25] (* _Jean-François Alcover_, Sep 12 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) = {2*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]-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. A053467, A003086, A053588.

%K nonn

%O 1,2

%A _Vladeta Jovovic_, Jan 21 2000

%E Terms a(14) and beyond from _Andrew Howroyd_, Sep 16 2018