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A220820 Number of rooted binary leaf-multilabeled trees with n leaves on the label set [3]. 2


%S 0,0,3,27,180,1089,6333,36309,207255,1184829,6799473,39224568,

%T 227554278,1327688328,7789644996,45944320449,272329169232,

%U 1621636729257,9697502473014,58219671592098,350791368690516,2120672113779573,12859687160772771,78201772222916649

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

%H Andrew Howroyd, <a href="/A220820/b220820.txt">Table of n, a(n) for n = 1..200</a>

%H 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 b:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,

%p (t-> t*(1-t)/2)(b(n/2, k)))+add(b(i, k)*b(n-i, k), i=1..n/2))

%p end:

%p a:= n-> (k-> add((-1)^i*binomial(k, i)*b(n, k-i), i=0..k))(3):

%p seq(a(n), n=1..30); # _Alois P. Heinz_, Sep 07 2019

%t 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 a[n_] := Function[k, Sum[(-1)^i Binomial[k, i] b[n, k - i], {i, 0, k}]][3];

%t Array[a, 30] (* _Jean-Fran├žois Alcover_, Apr 08 2020, after _Alois P. Heinz_ *)

%Y Column k=3 of A319541.

%K nonn

%O 1,3

%A _N. J. A. Sloane_, Dec 22 2012

%E Terms a(11) and beyond from _Andrew Howroyd_, Sep 23 2018

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Last modified September 20 03:49 EDT 2020. Contains 337264 sequences. (Running on oeis4.)