

A105821


Triangle of the numbers of different forests with one or more isolated vertices. Those forests have order N and m trees.


4



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, 69, 34, 16, 8, 4, 2, 1, 1
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OFFSET

1,12


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).


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 one or more parts equal to 1, of Product_{i=1..N} binomial(A000055(i)+Ki1, Ki).


EXAMPLE

a(5,2) = 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.
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;


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



