A319155 Number of bicolored graphs on 2n unlabeled nodes without isolated nodes and which are invariant when the two color classes are interchanged.

1, 1, 3, 11, 51, 337, 3500, 60936, 1866002, 102768062, 10296340496, 1890236147880, 639528747831552, 400813006079742544, 467517947968588109568, 1019290779610824185400096, 4170141472168738281510957264, 32130367702064742239376997422512

Offset: 0

Author

Andrew Howroyd, Sep 25 2018

Keywords

• nonn

Links

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

Crossrefs

Cf. A007140, A122082.

Programs

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];
A122082[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
a[n_] := A122082[n] - A122082[n-1];
a /@ Range[0, 17] (* Jean-François Alcover, Sep 05 2019, after Andrew Howroyd in A122082 *)

Formula

a(n) = A122082(n) - A122082(n-1).