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%I #7 May 09 2013 12:36:27
%S 1,6,42,380,4320,59682,974848,18423288,396000000,9548713790,
%T 255409127424,7507985556084,240659872940032,8355664160156250,
%U 312437224148828160,12519386633593104368,535233488907211702272,24320165501859426874998,1170472960000000000000000,59483046140261749951587180
%N Total number of rooted labeled trees over all forests on {1,2,...,n} in which one tree has been specially designated.
%C The expected number of trees in each forest approaches 5/2 as n gets large.
%F 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.
%F a(n) = Sum_{k=1..n} A225465(n,k)*k.
%e 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 '.
%e ...1'... ...2'... ...1'..2... ...1..2'...
%e ...| ... ...| ... ........... ...........
%e ...2 ... ...1 ... ........... ...........
%t Table[Sum[Binomial[n - 1, k - 1] n^(n - k) k^2, {k, 1, n}], {n, 1,
%t 20}]
%K nonn
%O 1,2
%A _Geoffrey Critzer_, May 08 2013
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