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

0, 0, 3, 27, 180, 1089, 6333, 36309, 207255, 1184829, 6799473, 39224568, 227554278, 1327688328, 7789644996, 45944320449, 272329169232, 1621636729257, 9697502473014, 58219671592098, 350791368690516, 2120672113779573, 12859687160772771, 78201772222916649

Offset: 1

Author

N. J. A. Sloane, Dec 22 2012

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.

Crossrefs

Column k=3 of A319541.

Programs

Maple

b:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(b(n/2, k)))+add(b(i, k)*b(n-i, k), i=1..n/2))
    end:
a:= n-> (k-> add((-1)^i*binomial(k, i)*b(n, k-i), i=0..k))(3):
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 07 2019

Mathematica

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}]];
a[n_] := Function[k, Sum[(-1)^i Binomial[k, i] b[n, k - i], {i, 0, k}]][3];
Array[a, 30] (* Jean-François Alcover, Apr 08 2020, after Alois P. Heinz *)

Extensions

Terms a(11) and beyond from Andrew Howroyd, Sep 23 2018