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A105821
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Triangle of the numbers of different forests with one or more isolated vertices. Those forests have order N and m trees.
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1
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1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 3, 3, 2, 1, 1, 0, 6, 6, 4, 2, 1, 1, 0, 11, 11, 7, 4, 2, 1, 1, 0, 23, 23, 14, 8, 4, 2, 1, 1, 0, 47, 46, 29, 15, 8, 4, 2, 1, 1, 0, 106, 99, 60, 32, 16, 8, 4, 2, 1, 1, 0, 235, 216, 128, 66, 33, 16, 8, 4, 2, 1, 1, 0, 551, 488, 284, 143
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,12
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COMMENTS
| The unique tree with an isolated node has order one. For N > 1 and m > 1 there is at least one partition of N in m parts, with a part equal to 1, so a(n) > 0, when m > 1 and a(n) = 0, when m = 1 and N > 1. A095133(n) = A105821(n) + A105820(n).
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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 one or more parts equal to 1, of product_{1=<i<=N}C(A000055[i]+Ki-1, Ki).
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EXAMPLE
| a(12) = 2 because 5 vertices can be partitioned in two trees only in one way: one tree gets 4 nodes and the other tree gets 1. Since A000055[4] = 2 and
A000055[1] = 1: there are 2 forests. The forests of order less than or equal to 5 are depicted in the Weisstein "Forest" link.
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CROSSREFS
| Cf. A095133, A105820.
Sequence in context: A061198 A195011 A039801 * A004564 A144157 A004562
Adjacent sequences: A105818 A105819 A105820 * A105822 A105823 A105824
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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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