|
|
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
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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.
|
|
LINKS
|
|
|
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.
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,
|
|
MATHEMATICA
|
Table[Table[Binomial[n - 1, k - 1] n^(n - k) k, {k, 1, n}], {n, 1,
8}] // Grid
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|