

A105820


Triangle giving the numbers of different forests of m trees of smallest order 2, i.e., without isolated vertices.


3



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, 1, 0, 0, 0, 0, 0, 0, 0
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OFFSET

1,7


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 giving only one vertex to a tree.


LINKS

Eric Weisstein's World of Mathematics, Forest


FORMULA

a(n) = sum over the partitions of N: 1K1 + 2K2 + ... + NKN, with exactly m parts and no part equal to 1, of Product_{i=1..N} binomial(A000055(i)+Ki1, Ki).


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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KEYWORD



AUTHOR



STATUS

approved



