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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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2
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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; refs; listen; history; internal format)
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OFFSET
| 1,7
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COMMENTS
| 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
| Eric Weisstein's World of Mathematics, Forest
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FORMULA
| 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.rs), Apr 27 2005
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EXAMPLE
| 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
| Cf. A033185, A105786.
Sequence in context: A132013 A128229 A145677 * A136263 A105593 A029371
Adjacent sequences: A105817 A105818 A105819 * A105821 A105822 A105823
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KEYWORD
| nonn,tabl
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AUTHOR
| Washington Bomfim (webonfim(AT)bol.com.br), Apr 25 2005
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