A283417 Number T(n,k) of triangle-free graphs on n unlabeled nodes with exactly k connected components; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 3, 2, 1, 1, 0, 6, 4, 2, 1, 1, 0, 19, 10, 5, 2, 1, 1, 0, 59, 28, 11, 5, 2, 1, 1, 0, 267, 90, 32, 12, 5, 2, 1, 1, 0, 1380, 363, 100, 33, 12, 5, 2, 1, 1, 0, 9832, 1784, 397, 104, 34, 12, 5, 2, 1, 1, 0, 90842, 11770, 1892, 407, 105, 34, 12, 5, 2, 1, 1

Offset: 0

Author

Alois P. Heinz, Apr 14 2017

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
  1;
  0,    1;
  0,    1,    1;
  0,    1,    1,   1;
  0,    3,    2,   1,   1;
  0,    6,    4,   2,   1,  1;
  0,   19,   10,   5,   2,  1,  1;
  0,   59,   28,  11,   5,  2,  1, 1;
  0,  267,   90,  32,  12,  5,  2, 1, 1;
  0, 1380,  363, 100,  33, 12,  5, 2, 1, 1;
  0, 9832, 1784, 397, 104, 34, 12, 5, 2, 1, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..17, flattened
• Index entries for triangles generated by the Multiset Transformation

Crossrefs

Columns k=0-1 give: A000007, A024607.

Row sums give A006785.

Formula

G.f.: Product_{j>=1} 1/(1-y*x^j)^A024607(j).