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

%I A122082 #21 Mar 11 2020 19:03:44
%S A122082 1,2,5,16,67,404,3904,64840,1930842,104698904,10401039400,
%T A122082 1900637187280,641429385018832,401454435464761376,
%U A122082 467919402404052870944,1019758699013228238271040,4171161230867751509749228304
%N A122082 Number of unlabeled bicolored graphs on 2n nodes which are invariant when the two color classes are interchanged.
%D A122082 R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
%H A122082 Andrew Howroyd, <a href="/A122082/b122082.txt">Table of n, a(n) for n = 0..50</a>
%H A122082 F. Harary, L. March and R. W. Robinson, <a href="https://doi.org/10.1068/b050031">On enumerating certain design problems in terms of bicolored graphs with no isolates</a>, Environment and Planning, B 5 (1978), 31-43.
%H A122082 F. Harary, L. March and R. W. Robinson, <a href="/A007139/a007139.pdf">On enumerating certain design problems in terms of bicolored graphs with no isolates</a>, Environment and Planning B: Urban Analytics and City Science, 5 (1978), 31-43. [Annotated scanned copy]
%F A122082 a(n) = 2*A007139(n) - A002724(n). - _Vladeta Jovovic_, Feb 27 2007
%t A122082 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 A122082 edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total @ Quotient[v + 1, 2]
%t A122082 a[n_] := (s=0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
%t A122082 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 06 2018, after _Andrew Howroyd_ *)
%o A122082 (PARI)
%o A122082 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 A122082 edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, (v[i]+1)\2)}
%o A122082 a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ _Andrew Howroyd_, Oct 23 2017
%Y A122082 Row sums of A123548.
%Y A122082 Cf. A002724, A007139, A333160.
%K A122082 nonn
%O A122082 0,2
%A A122082 _N. J. A. Sloane_, Oct 18 2006
%E A122082 More terms from _Vladeta Jovovic_, Feb 27 2007