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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 (list; table; graph; refs; listen; history; text; internal format)
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)+Ki-1, 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>=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,

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

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Last modified December 16 19:11 EST 2018. Contains 318188 sequences. (Running on oeis4.)