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A122082 Number of unlabeled bicolored graphs on 2n nodes which are invariant when the two color classes are interchanged. 5
1, 2, 5, 16, 67, 404, 3904, 64840, 1930842, 104698904, 10401039400, 1900637187280, 641429385018832, 401454435464761376, 467919402404052870944, 1019758699013228238271040, 4171161230867751509749228304 (list; graph; refs; listen; history; text; internal format)
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

Cf. A002724, A007139.

Sequence in context: A275518 A005163 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

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Last modified March 19 04:34 EDT 2019. Contains 321311 sequences. (Running on oeis4.)