A106240 Triangle read by rows: T(n,m) = number of unlabeled cographs on n nodes with m connected components.

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

Offset: 1

Author

Washington Bomfim, May 06 2005

Keywords

• nonn
• tabl

Examples

			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>=n-1. 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>=n-1 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,
		

Links

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

Crossrefs

Cf. A000669 (first column), A000084 (row sums), A201922.

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)+Ki-1, Ki).