A052303 Number of asymmetric Greg trees.

1, 1, 0, 0, 0, 0, 1, 4, 12, 42, 137, 452, 1491, 4994, 16831, 57408, 197400, 685008, 2395310, 8437830, 29917709, 106724174, 382807427, 1380058180, 4998370015, 18181067670, 66393725289, 243347195594, 894959868983, 3301849331598, 12217869541117, 45335177297876

Offset: 0

Author

Christian G. Bower, Nov 15 1999

Keywords

• nonn

Comments

A Greg tree can be described as a tree with 2-colored nodes where only the black nodes are counted and the white nodes are of degree at least 3.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1668
• Index entries for sequences related to trees

Crossrefs

Cf. A005263, A005264, A048159, A048160, A052300-A052302.

Programs

Maple

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

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[g[i], j] b[n - i j, i - 1], {j, 0, n/i}]]];
g[n_] := If[n < 1, 0, b[n - 1, n - 1] + b[n, n - 1]];
a[n_] := If[n == 0, 1, g[n] - Sum[g[j] g[n - j], {j, 0, n}]];
a /@ Range[0, 40] (* Jean-François Alcover, Apr 28 2020, after Alois P. Heinz *)

Formula

G.f.: 1+B(x)-B(x)^2 where B(x) is g.f. of A052301.