%I #9 Feb 16 2025 08:34:01

%S 1,1,2,4,11,29,91,299,1057,3884,14883,58508,235771,967790,4037807,

%T 17074475,73058753,315803342,1377445726,6056134719,26817483095,

%U 119516734167,535751271345,2414304071965,10932421750492,49723583969029,227079111492652,1040939109111200,4788357522831785

%N Number of bicolored trees on n unlabeled nodes such that every white node is adjacent to a black node.

%C 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 trees with n nodes.

%H Andrew Howroyd, <a href="/A339834/b339834.txt">Table of n, a(n) for n = 0..500</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominatingSet.html">Dominating Set</a>

%e a(2) = 2 because at most one node can be colored white.

%e a(3) = 4 because the only tree is the path graph P_3. If the center node is colored white then both of the ends must be colored black; otherwise zero, one or both of the ends can be colored black. In total there are 4 possibilities.

%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o seq(n)={my(u=v=w=[]); for(n=1, n, my(t1=EulerT(v), t2=EulerT(u+v)); u=concat([1], EulerT(u+v+w)); v=concat([0], t2-t1); w=concat([1], t1)); my(g=x*Ser(u+v), guw=x^2*Ser(u)*Ser(w)); Vec(1 + g + (subst(g,x,x^2) - g^2 - 2*guw)/2)}

%Y Cf. A038056 (bicolored trees), A339830, A339833, A339835 (rooted), A339836, A339837.

%K nonn

%O 0,3

%A _Andrew Howroyd_, Dec 19 2020