A108530 Number of rooted identity trees with n internal (non-leaf) nodes.

1, 1, 2, 4, 12, 34, 110, 364, 1248, 4356, 15520, 56022, 204726, 755472, 2812004, 10543718, 39791070, 151022006, 576090250, 2207493080, 8493196536, 32797115398, 127071214442, 493831241234, 1924504466246, 7519182311366, 29447430754182, 115577336981932

Offset: 0

Author

Christian G. Bower, Jun 07 2005

Keywords

• nonn
• eigen

Comments

Also for n>0, rooted trees with n nodes and 2-colored internal nodes. Black nodes correspond to nodes with a leaf child; white nodes correspond to those without one.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• N. J. A. Sloane, Transforms
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A004111, A004113, A108531, A108532.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i$2), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<2, 1, 2*b(n-1,n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, May 20 2013

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<2, 1, 2*b[n-1, n-1]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 01 2016, after Alois P. Heinz *)

Formula

Shifts left and halves under WEIGH transform.

a(n) ~ c * d^n / n^(3/2), where d = 4.1516890102085520777311008746639624... and c = 0.3329810927479684511418598248... - Vaclav Kotesovec, Feb 28 2014