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A105785
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Number of different forests of rooted trees, without isolated vertices, on n labeled nodes.
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0
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0, 2, 9, 76, 805, 10626, 167839, 3091768, 65127465, 1544951350, 40770052411, 1184951084340, 37616775522781, 1295202587597842, 48080003446006575, 1914305438178286576, 81379323738092982097, 3679128029385789284718
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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FORMULA
| a(n)= sum N/D over all the partitions of n:1K1+2K2+ ... + nKn, with smallest part greater than 1, where N = n!*product_{1=<i<=n}i^((i-1)Ki) and D = product_{1=<i<=n}(Ki!(i!)^Ki).
E.g.f.: -exp(-x)*LambertW(-x)/x. a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*(k+1)^(k-1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 22 2005
a(0) = 1, a(n) = Sum_{j=1..n-1} C(n-1,j) (j+1)^j a(n-1-j) if n>0. [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 15 2008]
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EXAMPLE
| a(5)= 805 because there are 625 such trees and 5 vertices can be partitioned in two trees only in one way: 3 go to one tree and 2 go to the other. It's impossible to split 5 vertices in 3 or more trees without give only one vertex to a tree. Each one of the 3^2 distinct trees on 3 vertices can be labeled in C(5, 3) manners and to each one of the 9C(5, 3) = 90 possibilities there are 2 different trees of order 2, so we get 180 forests of two trees.
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MAPLE
| a:= proc(n) option remember; if n=0 then 1 else add (binomial (n-1, j) *(j+1)^j *a(n-1-j), j=1..n-1) fi end: seq (a(n), n=1..25); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 15 2008]
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CROSSREFS
| Cf. A033185, A105599.
Sequence in context: A029849 A197082 A080638 * A123680 A132621 A108992
Adjacent sequences: A105782 A105783 A105784 * A105786 A105787 A105788
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KEYWORD
| nonn
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AUTHOR
| Washington Bomfim (webonfim(AT)bol.com.br), Apr 21 2005
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EXTENSIONS
| More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 15 2008
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