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A220820 Number of rooted binary leaf-multilabeled trees with n leaves on the label set [3].

%I A220820 #16 Apr 08 2020 07:33:00
%S A220820 0,0,3,27,180,1089,6333,36309,207255,1184829,6799473,39224568,
%T A220820 227554278,1327688328,7789644996,45944320449,272329169232,
%U A220820 1621636729257,9697502473014,58219671592098,350791368690516,2120672113779573,12859687160772771,78201772222916649
%N A220820 Number of rooted binary leaf-multilabeled trees with n leaves on the label set [3].
%H A220820 Andrew Howroyd, <a href="/A220820/b220820.txt">Table of n, a(n) for n = 1..200</a>
%H A220820 V. P. Johnson, <a href="http://people.math.sc.edu/czabarka/Theses/JohnsonThesis.pdf">Enumeration Results on Leaf Labeled Trees</a>, Ph. D. Dissertation, Univ. Southern Calif., 2012.
%p A220820 b:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
%p A220820       (t-> t*(1-t)/2)(b(n/2, k)))+add(b(i, k)*b(n-i, k), i=1..n/2))
%p A220820     end:
%p A220820 a:= n-> (k-> add((-1)^i*binomial(k, i)*b(n, k-i), i=0..k))(3):
%p A220820 seq(a(n), n=1..30);  # _Alois P. Heinz_, Sep 07 2019
%t A220820 b[n_, k_] := b[n, k] = If[n < 2, k n, If[OddQ[n], 0, Function[t, t(1 - t)/2 ][b[n/2, k]]] + Sum[b[i, k] b[n - i, k], {i, 1, n/2}]];
%t A220820 a[n_] := Function[k, Sum[(-1)^i Binomial[k, i] b[n, k - i], {i, 0, k}]][3];
%t A220820 Array[a, 30] (* _Jean-François Alcover_, Apr 08 2020, after _Alois P. Heinz_ *)
%Y A220820 Column k=3 of A319541.
%K A220820 nonn
%O A220820 1,3
%A A220820 _N. J. A. Sloane_, Dec 22 2012
%E A220820 Terms a(11) and beyond from _Andrew Howroyd_, Sep 23 2018