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A292668 Number of forests of exactly n (unlabeled) ordered rooted trees with a total of 2n non-root nodes. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Each tree has at least 1 non-root node.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1666

Index entries for sequences related to rooted trees

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

EXAMPLE

: 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   .     .      .       .       .       .         .         .

:

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);

CROSSREFS

Cf. A000108, A275431.

Sequence in context: A133592 A115967 A150714 * A122447 A150715 A026528

Adjacent sequences:  A292665 A292666 A292667 * A292669 A292670 A292671

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 20 2017

STATUS

approved

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Last modified December 8 01:45 EST 2019. Contains 329850 sequences. (Running on oeis4.)