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A339832 Number of bicolored graphs on n unlabeled nodes such that black nodes are not adjacent to each other.

%I A339832 #12 Feb 16 2025 08:34:01
%S A339832 1,2,5,14,50,230,1543,16252,294007,9598984,577914329,64384617634,
%T A339832 13264949930889,5055918209734322,3572106887472105263,
%U A339832 4692016570446185240464,11496632576435936553085113,52730955262459923752850296554,454273406825238417871411598421653
%N A339832 Number of bicolored graphs on n unlabeled nodes such that black nodes are not adjacent to each other.
%C A339832 The black nodes form an independent vertex set. For n > 0, a(n) is then the total number of indistinguishable independent vertex sets summed over distinct unlabeled graphs on n nodes.
%H A339832 Andrew Howroyd, <a href="/A339832/b339832.txt">Table of n, a(n) for n = 0..40</a>
%H A339832 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>
%o A339832 (PARI)
%o A339832 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 A339832 edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
%o A339832 cross(u, v) = {sum(i=1, #u, sum(j=1, #v, gcd(u[i], v[j])))}
%o A339832 U(nb,nw)={my(s=0); forpart(v=nw, my(t=0); forpart(u=nb, t += permcount(u) * 2^cross(u,v)); s += t*permcount(v) * 2^edges(v)/nb!); s/nw!}
%o A339832 a(n) = {sum(k=0, n, U(k, n-k))}
%Y A339832 A049312 counts bicolored graphs where adjacent nodes cannot have the same color.
%Y A339832 A000666 counts bicolored graphs where adjacent nodes can have the same color.
%Y A339832 Cf. A079491 (labeled case), A339830 (trees), A339836, A340021.
%K A339832 nonn
%O A339832 0,2
%A A339832 _Andrew Howroyd_, Dec 19 2020