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%I #16 Sep 22 2019 08:03:02
%S 1,1,3,16,121,1480,50993,3279685,505641590,152461906778,
%T 103587671805408,145528904385412088,442626996609870050404,
%U 2918362542591139744394993,40446812392580562094804791143,1260273961234324967695235253182680,80686628450087709982052029871655471264
%N Number of self-complementary 4-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="/A053588/b053588.txt">Table of n, a(n) for n = 1..50</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_] := Module[{m = 1, s = 0, t, i, k = 0}, 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[If[EvenQ[v[[i]]*v[[j]]], GCD[v[[i]], v[[j]]], 0], {i, 2, Length[v]}, {j, 1, i - 1}] + Sum[If[EvenQ[v[[i]]], 2 Quotient[v[[i]], 4], 0], {i, 1, Length[v]}];
%t a[n_] := Module[{s = 0}, Do[s += permcount[p]*5^edges[p], {p, IntegerPartitions[n]}]; s/n!];
%t a /@ Range[1, 20] (* _Jean-François Alcover_, Sep 22 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]\4*2))}
%o a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*5^edges(p)); s/n!} \\ _Andrew Howroyd_, Sep 17 2018
%Y Cf. A004102, A004104, A053420.
%K nonn
%O 1,3
%A _Vladeta Jovovic_, Jan 19 2000
%E Terms a(16) and beyond from _Andrew Howroyd_, Sep 17 2018
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