A292668 Number of forests of exactly n (unlabeled) ordered rooted trees with a total of 2n non-root nodes.

1, 2, 8, 28, 105, 384, 1442, 5388, 20317, 76712, 290790, 1104538, 4205909, 16044994, 61322356, 234739140, 899911685, 3454630372, 13278582906, 51098682962, 196853475135, 759139115962, 2930340545406, 11321631496180, 43779660235746, 169429224658130

Offset: 0

Author

Alois P. Heinz, Sep 20 2017

Keywords

• nonn

Comments

Each tree has at least 1 non-root node.

Examples

			: a(2) = 8: (2 trees in each forest having 4 non-root nodes)
:
: o o . o o .  o o .  o  o .  o  o . o  o  .   o   o .  o   o  .
: | | . | | .  | | . ( ) | . ( ) | . | ( ) .  /|\  | . ( ) ( ) .
: o o . o o .  o o . o o o . o o o . o o o . o o o o . o o o o .
: |   . | | . ( )  . |     .   |   . |     .         .         .
: o   . o o . o o  . o     .   o   . o     .         .         .
: |   .     .      .       .       .       .         .         .
: o   .     .      .       .       .       .         .         .
:
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1666
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A000108, A275431.

Programs

Maple

C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(C(d+1)
      *d, d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);

Mathematica

c[n_] := c[n] = Binomial[2n, n]/(n+1);
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[c[d+1] d, {d, Divisors[j]}] a[n-j], {j, 1, n}]/n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)

Formula

G.f.: Product_{j>=1} 1/(1-x^j)^A000108(j+1).

a(n) = A275431(2n,n).

a(n) ~ c * 4^n / n^(3/2), where c = 49.48222899350915021666300344559315... - Vaclav Kotesovec, Sep 27 2017