A052300 Number of rooted Greg trees.

1, 2, 6, 21, 78, 313, 1306, 5653, 25088, 113685, 523522, 2443590, 11533010, 54949539, 263933658, 1276652682, 6213207330, 30402727854, 149486487326, 738184395770, 3659440942282, 18205043615467, 90856842218506, 454770531433586, 2282393627458496, 11483114908752959

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
• Index entries for sequences related to rooted trees
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A005263, A005264, A048159, A048160, A052301-A052303.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<1, 0, b(n-1$2)+b(n, n-1)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jun 22 2018

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i] + j - 1, j] b[n - i j, i - 1], {j, 0, n/i}]]];
a[n_] := If[n < 1, 0, b[n - 1, n - 1] + b[n, n - 1]];
a /@ Range[1, 40] (* Jean-François Alcover, Oct 02 2019, after Alois P. Heinz *)

Formula

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

a(n) ~ c * d^n / n^(3/2), where d = 5.33997181362574740496306748840603859910694551382103293340704... and c = 0.18146848896221859476228524468003196434835879494225205... - Vaclav Kotesovec, Jun 11 2021