OFFSET
0,3
COMMENTS
The black nodes form a dominating set. For n > 0, a(n) is then the total number of indistinguishable dominating sets summed over distinct unlabeled graphs on n nodes.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..40
Eric Weisstein's World of Mathematics, Dominating Set
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) * 2^edges(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 19 2020
STATUS
approved