A122087 Triangle read by rows: T(n,k) = number of unlabeled free bicolored trees with n nodes (n >= 1) and k (1 <= k <= floor(n/2), except k = 0 if n = 1 ) nodes of one color and n-k nodes of the other color (the colors are interchangeable).

1, 1, 1, 1, 1, 1, 2, 1, 2, 3, 1, 3, 7, 1, 3, 10, 9, 1, 4, 14, 28, 1, 4, 19, 45, 37, 1, 5, 24, 73, 132, 1, 5, 30, 105, 242, 168, 1, 6, 37, 152, 412, 693, 1, 6, 44, 204, 660, 1349, 895, 1, 7, 52, 274, 1008, 2472, 3927, 1, 7, 61, 351, 1479, 4219, 8105, 5097, 1, 8

Offset: 1

Author

N. J. A. Sloane, Oct 19 2006

Keywords

• nonn
• tabf

Examples

			K M N gives the number N of unlabeled free bicolored trees with K nodes of one color and M nodes of the other color.
0 1 1
Total( 1) = 1
1 1 1
Total( 2) = 1
1 2 1
Total( 3) = 1
1 3 1
2 2 1
Total( 4) = 2
1 4 1
2 3 2
Total( 5) = 3
1 5 1
2 4 2
3 3 3
Total( 6) = 6
1 6 1
2 5 3
3 4 7
Total( 7) = 11
1 7 1
2 6 3
3 5 10
4 4 9
Total( 8) = 23
From _Andrew Howroyd_, Apr 05 2023: (Start)
Triangle begins:
  n\k| 0 1  2   3    4    5    6
 ----+----------------------------
   1 | 1;
   2 | . 1;
   3 | . 1;
   4 | . 1, 1;
   5 | . 1, 2;
   6 | . 1, 2,  3;
   7 | . 1, 3,  7;
   8 | . 1, 3, 10,   9;
   9 | . 1, 4, 14,  28;
  10 | . 1, 4, 19,  45,  37;
  11 | . 1, 5, 24,  73, 132;
  12 | . 1, 5, 30, 105, 242, 168;
    ...
(End)
		

References

• R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.

Links

• R. W. Robinson, Rows 1 through 30, flattened

Crossrefs

Row sums give A000055.

Cf. A119856, A329054, A362019 (labeled version).

Formula

T(n,k) = A329054(k, n-k) for 2*k < n; T(2*n,n) = A119856(n). - Andrew Howroyd, Apr 04 2023