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A106236
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Triangle of the numbers of different forests with m rooted trees having distinct orders.
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1
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1, 1, 0, 2, 1, 0, 4, 2, 0, 0, 9, 6, 0, 0, 0, 20, 13, 2, 0, 0, 0, 48, 37, 4, 0, 0, 0, 0, 115, 86, 17, 0, 0, 0, 0, 0, 286, 239, 46, 0, 0, 0, 0, 0, 0, 719, 577, 142, 8, 0, 0, 0, 0, 0, 0, 1842, 1607, 367, 18, 0, 0, 0, 0, 0, 0, 0, 4766, 4025, 1136, 76, 0, 0, 0, 0, 0, 0, 0, 0, 12486, 11185
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OFFSET
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1,4
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COMMENTS
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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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LINKS
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Table of n, a(n) for n=1..80.
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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(A000081(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}A000081(i). (Naturally we do not consider the parts with multiplicity 0).
G.f.: Product_{k>0} (1+y*A000081(k)*x^k). - Vladeta Jovovic, May 14 2005
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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. A106234, A000081.
Sequence in context: A166555 A136329 A122073 * A122792 A139136 A138002
Adjacent sequences: A106233 A106234 A106235 * A106237 A106238 A106239
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KEYWORD
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nonn,tabl,changed
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AUTHOR
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Washington Bomfim, Apr 28 2005
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STATUS
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approved
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