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A225497 Total number of rooted labeled trees over all forests on {1,2,...,n} in which one tree has been specially designated. 2
1, 6, 42, 380, 4320, 59682, 974848, 18423288, 396000000, 9548713790, 255409127424, 7507985556084, 240659872940032, 8355664160156250, 312437224148828160, 12519386633593104368, 535233488907211702272, 24320165501859426874998, 1170472960000000000000000, 59483046140261749951587180 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for 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.

EXAMPLE

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 ...   ...........   ...........

MATHEMATICA

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

  20}]

CROSSREFS

Sequence in context: A052589 A074107 A187121 * A336950 A304071 A052608

Adjacent sequences:  A225494 A225495 A225496 * A225498 A225499 A225500

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, May 08 2013

STATUS

approved

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Last modified August 14 05:20 EDT 2022. Contains 356110 sequences. (Running on oeis4.)