OFFSET
0,3
COMMENTS
The black nodes form a maximal independent vertex set (or a set that is both independent and dominating). For n > 0, a(n) is then the total number of indistinguishable maximal independent vertex sets summed over distinct unlabeled graphs with n nodes.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..40
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set
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, 2]];
dom[u_, v_] := Product[2^Sum[GCD[u[[i]], v[[j]]], {j, 1, Length[v]}] - 1, {i, 1, Length[u]}];
U[nb_, nw_] := Module[{s = 0}, Do[t = 0; Do[t += permcount[v]*dom[u, v], {v, IntegerPartitions[nb]}]; s += t*permcount[u]*2^edges[u]/nb!, {u, IntegerPartitions[nw]}]; s/nw!];
a[n_] := Sum[U[k, n - k], {k, 0, n}];
Array[a, 20] (* Jean-François Alcover, Jan 07 2021, 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]\2)}
dom(u, v) = {prod(i=1, #u, 2^sum(j=1, #v, gcd(u[i], v[j]))-1)}
U(nb, nw)={my(s=0); forpart(u=nw, my(t=0); forpart(v=nb, t += permcount(v) * dom(u, v)); s += t*permcount(u) * 2^edges(u)/nb!); s/nw!}
a(n)={sum(k=0, n, U(k, n-k))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Dec 30 2020
STATUS
approved