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Search: id:A106235
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| A106235 |
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Triangle of the numbers of different forests of m rooted trees of smallest order 2, i.e., without isolated vertices. |
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1
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| 0, 1, 0, 2, 0, 0, 4, 1, 0, 0, 9, 2, 0, 0, 0, 20, 7, 1, 0, 0, 0, 48, 17, 2, 0, 0, 0, 0, 115, 48, 7, 1, 0, 0, 0, 0, 286, 124, 21, 2, 0, 0, 0, 0, 0, 719, 336, 60, 7, 1, 0, 0, 0, 0, 0, 1842, 888, 171, 21, 2, 0, 0, 0, 0, 0, 0, 4766, 2393, 488, 65, 7, 1, 0, 0
(list; table; graph; listen)
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OFFSET
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1,4
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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. A033185(n) = A106235(n) + A106234(n).
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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(A000081(i)+Ki-1, Ki).
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EXAMPLE
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a(12)=2 because 5 nodes can be partitioned in two trees only in a way: one tree gets 3 nodes and the other tree gets 2. Since A000081(3) = 2 and A000081(2)=1, there are two forests.
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CROSSREFS
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Cf. A033185, A106234.
Sequence in context: A061669 A136334 A155039 this_sequence A118965 A121552 A158118
Adjacent sequences: A106232 A106233 A106234 this_sequence A106236 A106237 A106238
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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 26 2005
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