

A106240


Triangle read by rows: T(n,m) = number of unlabeled cographs on n nodes with m connected components.


9



1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 12, 7, 3, 1, 1, 33, 20, 8, 3, 1, 1, 90, 55, 22, 8, 3, 1, 1, 261, 162, 63, 23, 8, 3, 1, 1, 766, 477, 188, 65, 23, 8, 3, 1, 1, 2312, 1450, 564, 196, 66, 23, 8, 3, 1, 1, 7068, 4446, 1732, 590, 198, 66, 23, 8, 3, 1, 1, 21965, 13858, 5384, 1824, 598, 199, 66, 23, 8, 3, 1, 1
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OFFSET

1,4


LINKS

Alois P. Heinz, Rows n = 1..141, flattened
Washington Bomfim, Illustration of this sequence


FORMULA

T(n, m) = sum over the partitions of n with m parts: 1K1 + 2K2 + ... + nKn = n, K1 + K2 + ... + Kn = m, of Product_{i=1..n} binomial(A000669(i)+Ki1, Ki).


EXAMPLE

T(10,8) = 3 because the partitions of 10 with 8 parts are 31111111 and 22111111. The partition 31111111 corresponds to 2 graphs and the partition 22111111 corresponds to only one.
T(n,m) = 1, if and only if m>=n1. Because A000669(1)=A000669(2)=1, the partitions of n with all parts <=2 correspond to summands = 1. If there is only a summand (or partition), the total is equal to 1. It is clear that for m>=n1 there is only one partition of n with exactly m parts.
Triangle begins:
1,
1, 1,
2, 1, 1,
5, 3, 1, 1,
12, 7, 3, 1, 1,
33, 20, 8, 3, 1, 1,
90, 55, 22, 8, 3, 1, 1,


CROSSREFS

Cf. A000669 (first column), A000084 (row sums), A201922.
Sequence in context: A047884 A124328 A055818 * A097615 A288386 A062993
Adjacent sequences: A106237 A106238 A106239 * A106241 A106242 A106243


KEYWORD

nonn,tabl


AUTHOR

Washington Bomfim, May 06 2005


STATUS

approved



