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Search: id:A106237
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| A106237 |
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Triangle of the numbers of different forests with m trees having distinct orders. |
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+0
2
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| 1, 1, 0, 1, 1, 0, 2, 1, 0, 0, 3, 3, 0, 0, 0, 6, 5, 1, 0, 0, 0, 11, 11, 2, 0, 0, 0, 0, 23, 20, 5, 0, 0, 0, 0, 0, 47, 46, 11, 0, 0, 0, 0, 0, 0, 106, 93, 26, 2, 0, 0, 0, 0, 0, 0, 235, 216, 58, 3, 0, 0, 0, 0, 0, 0, 0, 551, 467, 139, 12, 0, 0, 0, 0, 0, 0, 0, 0, 1301, 1121, 307, 29, 0, 0, 0, 0, 0, 0, 0
(list; table; graph; listen)
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OFFSET
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1,7
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COMMENT
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a(n) = 0 if and only if n < m + (((1+m)*m - 1)^2 -1)/8, where m is the number of trees in the forests counted by a(n).
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FORMULA
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a(n)= sum over the partitions of N:1K1+2K2+ ... +NKN, with exactly m distinct parts, of product_{1=<i<=N}C(A000055(i)+Ki-1, Ki). Because all the multiplicities of the parts of the considered partitions are 1, or 0, we can simplify the formula to a(n)= sum over the partitions of N with exactly m distinct parts, of product_{1=<i<=N}A000055(i). (Naturally we do not consider the parts with multiplicity 0).
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EXAMPLE
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a(3)=0 because m = 2 and (see comments) 3 < (2 + 3).
a(4)>0 because m = 1. Note that (((1+m)*m - 1)^2 -1)/8 = 0, if m = 1. It is clear that n >= m.
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CROSSREFS
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Cf. A106236, A000055.
Sequence in context: A037854 A091229 A055334 this_sequence A071675 A034365 A103778
Adjacent sequences: A106234 A106235 A106236 this_sequence A106238 A106239 A106240
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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 28 2005
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