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Search: id:A105784
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| A105784 |
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Number of different forests of unrooted trees, without isolated vertices, on n labeled nodes. |
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+0
1
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| 0, 1, 3, 19, 155, 1641, 21427, 334377, 6085683, 126745435, 2975448641, 77779634571, 2241339267037, 70604384569005, 2414086713172695, 89049201691604881, 3525160713653081279
(list; graph; listen)
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OFFSET
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1,3
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FORMULA
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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-2)Ki) and D = product_{1=<i<=n}(Ki!(i!)^Ki).
Inverse binomial transform of A001858. E.g.f.: exp(-x-LambertW(-x)-LambertW(-x)^2/2). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 22 2005
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EXAMPLE
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a(4) = 19 because there are 19 different such forests on 4 labeled nodes: 4^2 are trees, 3 have two trees and none can have more than two trees.
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MAPLE
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b:= n->add (add (binomial(m, j) *binomial(n-1, n-m-j) *n^(n-m-j) *(m+j)!/ (-2)^j, j=0..m)/m!, m=0..n): a:= n-> add (b(k) *(-1)^(n-k) *binomial(n, k), k=0..n): seq (a(n), n=1..17); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 10 2008]
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CROSSREFS
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Cf. A033185, A105599.
Sequence in context: A007112 A007111 A113013 this_sequence A077046 A054765 A057719
Adjacent sequences: A105781 A105782 A105783 this_sequence A105785 A105786 A105787
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KEYWORD
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nonn
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AUTHOR
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Washington Bomfim (webonfim(AT)bol.com.br), Apr 21 2005
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