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Number of unlabeled bicolored graphs on 2n nodes which are invariant when the two color classes are interchanged.
8

%I #21 Mar 11 2020 19:03:44

%S 1,2,5,16,67,404,3904,64840,1930842,104698904,10401039400,

%T 1900637187280,641429385018832,401454435464761376,

%U 467919402404052870944,1019758699013228238271040,4171161230867751509749228304

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

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

%H Andrew Howroyd, <a href="/A122082/b122082.txt">Table of n, a(n) for n = 0..50</a>

%H 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 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 a(n) = 2*A007139(n) - A002724(n). - _Vladeta Jovovic_, Feb 27 2007

%t 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 edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total @ Quotient[v + 1, 2]

%t a[n_] := (s=0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 06 2018, after _Andrew Howroyd_ *)

%o (PARI)

%o 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 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 a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ _Andrew Howroyd_, Oct 23 2017

%Y Row sums of A123548.

%Y Cf. A002724, A007139, A333160.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Oct 18 2006

%E More terms from _Vladeta Jovovic_, Feb 27 2007