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Search: id:A105820
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| A105820 |
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Triangle giving the numbers of different forests of m trees of smallest order 2, i.e. without isolated vertices. |
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3
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| 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 3, 1, 0, 0, 0, 6, 3, 1, 0, 0, 0, 11, 5, 1, 0, 0, 0, 0, 23, 12, 3, 1, 0, 0, 0, 0, 47, 23, 6, 1, 0, 0, 0, 0, 0, 106, 52, 14, 3, 1, 0, 0, 0, 0, 0, 235, 110, 29, 6, 1, 0, 0, 0, 0, 0, 0, 551, 253, 68, 15, 3, 1, 0, 0, 0, 0, 0, 0, 1301, 570, 148, 31, 6
(list; table; graph; listen)
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OFFSET
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1,7
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COMMENT
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Forests of order N with m components, m > floor(N/2) must contain an isolated vertex since it is impossible to partition N vertices in floor(N/2) + 1 or more trees without give only one vertex to a tree.
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LINKS
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Eric Weisstein's World of Mathematics, Forest
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FORMULA
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a(n)= sum over the partitions of N:1K1+2K2+ ... +NKN, with exactly m parts, and no part equal to 1, of product_{1=<i<=N}C(A000055[i]+Ki-1, Ki).
G.f.: 1/Product((1-x*y^i)^A000055(i), i=2..infinity). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 27 2005
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EXAMPLE
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a(12)=1 because 5 nodes can be partitioned into two trees only in one way: one tree gets 3 nodes, and the other tree gets 2. Since A000055[3] = A000055[2]=1, there is only one forest. (The forests of order less than or equal to 5 are depicted in the Weisstein link).
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CROSSREFS
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Cf. A033185, A105786.
Sequence in context: A063658 A132013 A128229 this_sequence A136263 A105593 A029371
Adjacent sequences: A105817 A105818 A105819 this_sequence A105821 A105822 A105823
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KEYWORD
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nonn,tabl
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AUTHOR
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Washington Bomfim (webonfim(AT)bol.com.br), Apr 25 2005
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