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%I #13 May 11 2013 02:07:11
%S 1,2,2,9,12,3,64,96,36,4,625,1000,450,80,5,7776,12960,6480,1440,150,6,
%T 117649,201684,108045,27440,3675,252,7,2097152,3670016,2064384,573440,
%U 89600,8064,392,8
%N 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.
%C Row sums = 2n*(n+1)^(n-2) = A089946(offset).
%C The average number of trees in each forest approaches 5/2 as n gets large.
%F T(n,k) = binomial(n-1,k-1)*n^(n-k)*k = A061356(n,k)*k(offset).
%F E.g.f.: y*A(x)*exp(y*A(x)) where A(x) is e.g.f. for A000169.
%e T(2,1)=2 T(2,2)=2
%e ...1'... ...2'... ...1'..2... ...1..2'...
%e ...| ... ...| ... ........... ...........
%e ...2 ... ...1 ... ........... ...........
%e The root node is on top. The ' indicates the tree which has been specially designated.
%e 1,
%e 2, 2,
%e 9, 12, 3,
%e 64, 96, 36, 4,
%e 625, 1000, 450, 80, 5,
%e 7776, 12960, 6480, 1440, 150, 6,
%e 117649, 201684, 108045, 27440, 3675, 252, 7,
%t Table[Table[Binomial[n - 1, k - 1] n^(n - k) k, {k, 1, n}], {n, 1,
%t 8}] // Grid
%K nonn,tabl
%O 1,2
%A _Geoffrey Critzer_, May 08 2013
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