A052301 Number of asymmetric rooted Greg trees.

1, 1, 2, 5, 14, 43, 138, 455, 1540, 5305, 18546, 65616, 234546, 845683, 3072350, 11235393, 41326470, 152793376, 567518950, 2116666670, 7924062430, 29765741831, 112157686170, 423809991041, 1605622028100, 6097575361683, 23207825593664, 88512641860558

Offset: 1

Author

Christian G. Bower, Nov 15 1999

Keywords

• nonn
• eigen

Comments

A rooted Greg tree can be described as a rooted tree with 2-colored nodes where only the black nodes are counted and the white nodes have at least 2 children.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• N. J. A. Sloane, Transforms
• Index entries for sequences related to rooted trees

Crossrefs

Essentially the same as A031148. Cf. A005263, A005264, A048159, A048160, A052300-A052303.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<1, 1, b(n-1$2)) +b(n, n-1):
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 06 2014

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[a[i], j]*b[n - i*j, i-1], {j, 0, n/i}]]];
a[n_] := If[n<1, 1, b[n-1, n-1]] + b[n, n-1];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 01 2016, after Alois P. Heinz *)

Formula

Satisfies a = WEIGH(a) + SHIFT_RIGHT(WEIGH(a)) - a.

a(n) ~ c * d^n / n^(3/2), where d = 4.0278584853545190803008179085023154..., c = 0.14959176868229550510957320468... . - Vaclav Kotesovec, Sep 12 2014