A225497 Total number of rooted labeled trees over all forests on {1,2,...,n} in which one tree has been specially designated.

1, 6, 42, 380, 4320, 59682, 974848, 18423288, 396000000, 9548713790, 255409127424, 7507985556084, 240659872940032, 8355664160156250, 312437224148828160, 12519386633593104368, 535233488907211702272, 24320165501859426874998, 1170472960000000000000000, 59483046140261749951587180

Offset: 1

Author

Geoffrey Critzer, May 08 2013

Keywords

• nonn

Comments

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

Examples

			a(2) = 6 because there are 6 trees in these forests on 2 nodes.  The root node is on top and the designated tree is marked by '.
...1'...   ...2'...   ...1'..2...   ...1..2'...
...| ...   ...| ...   ...........   ...........
...2 ...   ...1 ...   ...........   ...........
		

Programs

Mathematica

Table[Sum[Binomial[n - 1, k - 1] n^(n - k) k^2, {k, 1, n}], {n, 1,
  20}]

Formula

a(n) = Sum_{k=1..n} binomial(n,k)*n^(n-k)*k^2 = ((1 + 1/n)^n n^(1 + n) (-1 + 5 n))/(1 + n)^3.

a(n) = Sum_{k=1..n} A225465(n,k)*k.