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%I #15 Sep 12 2019 08:47:37
%S 1,0,0,4,16,0,0,2080,32896,0,0,178990080,11453771776,0,0,
%T 3002404080455680,768614611824951296,0,0,10316167090130469587779584,
%U 10563755026498136326181748736,0,0,7561830376433501721102295492903043072,30973257220603971305905396442627825467392
%N Number of self-complementary 3-multigraphs on n nodes.
%D V. Jovovic, On the number of m-place relations (in Russian), Logiko-algebraicheskie konstruktsii, Tver, 1992, 59-66.
%D J. Xu, Ch. R. Wang, J. F. Wang, The theory of self-complementary k-multigraphs (in Chinese), Pure Appl. Math. [Chuncui Shuxue yu Yingyong Shuxue] 10 (1994), Special Issue, 18-22.
%H Andrew Howroyd, <a href="/A052107/b052107.txt">Table of n, a(n) for n = 1..100</a>
%H D. Wille, <a href="https://doi.org/10.1016/0095-8956(78)90034-5">Enumeration of self-complementary structures</a>, J. Comb. Theory B 25 (1978) 143-150.
%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] := 4 Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + 2 Total[v];
%t a[n_] := Module[{s = 0}, If[Mod[n, 4] < 2, Do[s += permcount[4*p]* 4^edges[p]*If[OddQ[n], n*4^Length[p], 1], {p, IntegerPartitions[ Quotient[n, 4]]}]]; 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) = {4*sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, 2*v[i])}
%o a(n) = {my(s=0); if(n%4<2, forpart(p=n\4, s+=permcount(4*Vec(p))*4^edges(p)*if(n%2, n*4^#p, 1))); s/n!} \\ _Andrew Howroyd_, Sep 17 2018
%Y Cf. A000171, A053588, A053400.
%K nonn
%O 1,4
%A _Vladeta Jovovic_, Jan 20 2000
%E Terms a(24) and beyond from _Andrew Howroyd_, Sep 17 2018
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