A122082 Number of unlabeled bicolored graphs on 2n nodes which are invariant when the two color classes are interchanged.

1, 2, 5, 16, 67, 404, 3904, 64840, 1930842, 104698904, 10401039400, 1900637187280, 641429385018832, 401454435464761376, 467919402404052870944, 1019758699013228238271040, 4171161230867751509749228304

Offset

0,2

References

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

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50

F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning, B 5 (1978), 31-43.

F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning B: Urban Analytics and City Science, 5 (1978), 31-43. [Annotated scanned copy]

Formula

a(n) = 2*A007139(n) - A002724(n). - Vladeta Jovovic, Feb 27 2007

Mathematica

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];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total @ Quotient[v + 1, 2]
a[n_] := (s=0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 06 2018, 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) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, (v[i]+1)\2)}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ Andrew Howroyd, Oct 23 2017

Crossrefs

Row sums of A123548.

Cf. A002724, A007139, A333160.

Sequence in context: A005163 A359986 A006116 * A002631 A107948 A220840

Adjacent sequences: A122079 A122080 A122081 * A122083 A122084 A122085

Keyword

nonn

Author

N. J. A. Sloane, Oct 18 2006

Extensions

More terms from Vladeta Jovovic, Feb 27 2007

Status

approved