A220819 Number of rooted binary leaf-multilabeled trees with n leaves on the label set [2].

0, 1, 4, 14, 48, 171, 614, 2270, 8518, 32567, 126168, 495079, 1962752, 7853581, 31672502, 128622480, 525523990, 2158818376, 8911039462, 36941520279, 153740822408, 642085403709, 2690217364606, 11304538078369, 47630350694248, 201181246749072, 851690546714230

Offset: 1

Author

N. J. A. Sloane, Dec 22 2012

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1542 (first 200 terms from Andrew Howroyd)
• V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.

Crossrefs

Column k=2 of A319541.

Cf. A001190, A083563.

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-> b(n, 2)-2*b(n, 1):
seq(a(n), n=1..40);  # Alois P. Heinz, Sep 07 2019

Mathematica

(* b = A083563 *) b[n_] := b[n] = If[n < 2, 2*n, If[OddQ[n], 0, #*(1 - #)/2 &[b[n/2]]]] + Sum[b[i]*b[n - i], {i, 1, n/2}];
(* c = A001190 *) c[n_?OddQ] := c[n] = Sum[c[k]*c[n - k], {k, 1, (n - 1)/2}]; c[n_?EvenQ] := c[n] = Sum[c[k]*c[n - k], {k, 1, n/2 - 1}] + (1/2)*c[n/2]*(1 + c[n/2]); c[0] = 0; c[1] = 1;
a[n_] := b[n] - 2 c[n];
Array[a, 27] (* Jean-François Alcover, Sep 07 2019 *)

Formula

a(n) = A083563(n) - 2*A001190(n). - Andrew Howroyd, Sep 23 2018

Extensions

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