A052107 Number of self-complementary 3-multigraphs on n nodes.

1, 0, 0, 4, 16, 0, 0, 2080, 32896, 0, 0, 178990080, 11453771776, 0, 0, 3002404080455680, 768614611824951296, 0, 0, 10316167090130469587779584, 10563755026498136326181748736, 0, 0, 7561830376433501721102295492903043072, 30973257220603971305905396442627825467392

Offset

1,4

References

V. Jovovic, On the number of m-place relations (in Russian), Logiko-algebraicheskie konstruktsii, Tver, 1992, 59-66.

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..100

D. Wille, Enumeration of self-complementary structures, J. Comb. Theory B 25 (1978) 143-150.

Mathematica

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];
edges[v_List] := 4 Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + 2 Total[v];
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!];
Array[a, 25] (* Jean-François Alcover, Sep 12 2019, after Andrew Howroyd *)

Prog

(PARI)
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}
edges(v) = {4*sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, 2*v[i])}
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

Crossrefs

Cf. A000171, A053588, A053400.

Sequence in context: A220788 A330315 A061849 * A069019 A188249 A228559

Adjacent sequences: A052104 A052105 A052106 * A052108 A052109 A052110

Keyword

nonn

Author

Vladeta Jovovic, Jan 20 2000

Extensions

Terms a(24) and beyond from Andrew Howroyd, Sep 17 2018

Status

approved