A225465 Triangular array read by rows: T(n, k) is the number of rooted forests on {1, 2, ..., n} in which one tree has been specially designated that contain exactly k trees; n >= 1, 1 <= k <= n.

1, 2, 2, 9, 12, 3, 64, 96, 36, 4, 625, 1000, 450, 80, 5, 7776, 12960, 6480, 1440, 150, 6, 117649, 201684, 108045, 27440, 3675, 252, 7, 2097152, 3670016, 2064384, 573440, 89600, 8064, 392, 8, 43046721, 76527504, 44641044, 13226976, 2296350, 244944, 15876, 576, 9

Offset

1,2

Comments

Row sums = 2n*(n+1)^(n-2) = A089946(offset).

The average number of trees in each forest approaches 5/2 as n gets large.

The rows give the coefficients of the derivatives of the Abel polynomials. - Peter Luschny, Feb 22 2025

Links

Table of n, a(n) for n=1..45.

Formula

T(n, k) = binomial(n-1, k-1)*n^(n-k)*k = A061356(n, k)*k(offset).

E.g.f.: y*A(x)*exp(y*A(x)) where A(x) is e.g.f. for A000169.

Example

    T(2,1)=2                  T(2,2)=2
  ...1'...   ...2'...   ...1'..2...   ...1..2'...
  ...| ...   ...| ...   ...........   ...........
  ...2 ...   ...1 ...   ...........   ...........
The root node is on top.  The ' indicates the tree which has been specially designated.
Triangle starts:
  [1]        1;
  [2]        2,        2;
  [3]        9,       12,        3;
  [4]       64,       96,       36,        4;
  [5]      625,     1000,      450,       80,       5;
  [6]     7776,    12960,     6480,     1440,     150,      6;
  [7]   117649,   201684,   108045,    27440,    3675,    252,     7;
  [8]  2097152,  3670016,  2064384,   573440,   89600,   8064,   392,   8;
  [9] 43046721, 76527504, 44641044, 13226976, 2296350, 244944, 15876, 576, 9;

Mathematica

Table[Table[Binomial[n - 1, k - 1] n^(n - k) k, {k, 1, n}], {n, 1, 8}] // Grid

Crossrefs

Cf. A061356, A089946 (row sums), A000169, A137452.

Sequence in context: A374303 A372641 A002880 * A387003 A248665 A066324

Adjacent sequences: A225462 A225463 A225464 * A225466 A225467 A225468

Keyword

nonn,tabl

Author

Geoffrey Critzer, May 08 2013

Status

approved