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 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 Sequence in context: A143022 A154100 A002880 * A248665 A066324 A143146 Adjacent sequences:  A225462 A225463 A225464 * A225466 A225467 A225468 KEYWORD nonn,tabl AUTHOR Geoffrey Critzer, May 08 2013 STATUS approved

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Last modified April 17 11:48 EDT 2021. Contains 343064 sequences. (Running on oeis4.)