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

%I A319155 #23 May 06 2025 11:14:36
%S A319155 1,1,3,11,51,337,3500,60936,1866002,102768062,10296340496,
%T A319155 1890236147880,639528747831552,400813006079742544,
%U A319155 467517947968588109568,1019290779610824185400096,4170141472168738281510957264,32130367702064742239376997422512
%N A319155 Number of bicolored graphs on 2n unlabeled nodes without isolated nodes and which are invariant when the two color classes are interchanged.
%H A319155 Andrew Howroyd, <a href="/A319155/b319155.txt">Table of n, a(n) for n = 0..50</a>
%F A319155 a(n) = A122082(n) - A122082(n-1).
%t A319155 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];
%t A319155 edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total @ Quotient[v + 1, 2];
%t A319155 A122082[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
%t A319155 a[n_] := A122082[n] - A122082[n-1];
%t A319155 a /@ Range[0, 17] (* _Jean-François Alcover_, Sep 05 2019, after _Andrew Howroyd_ in A122082 *)
%Y A319155 Cf. A007140, A122082.
%K A319155 nonn
%O A319155 0,3
%A A319155 _Andrew Howroyd_, Sep 25 2018