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Number of rooted bicolored trees on n unlabeled nodes such that black nodes are not adjacent to each other and every white node is adjacent to a black node.
4

%I #8 Jan 09 2021 22:10:51

%S 1,2,4,10,27,75,221,662,2042,6402,20407,65828,214720,706600,2343767,

%T 7826752,26293468,88796471,301290197,1026595232,3511246069,

%U 12050780294,41488523002,143246116231,495881545520,1720771421470,5984652387281,20857113949868,72829214554641,254762923125929

%N Number of rooted bicolored trees on n unlabeled nodes such that black nodes are not adjacent to each other and every white node is adjacent to a black node.

%H Andrew Howroyd, <a href="/A339838/b339838.txt">Table of n, a(n) for n = 1..500</a>

%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(v+w)); v=concat([0], t2-t1); w=concat([1], t1)); u+v}

%Y Cf. A038055 (rooted bicolored trees), A339831, A339835, A339837.

%K nonn

%O 1,2

%A _Andrew Howroyd_, Dec 20 2020