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

1,7

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

Formula

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

Example

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)

Crossrefs

Row sums give A000055.

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

Sequence in context: A144215 A254539 A283827 * A139642 A264744 A354322

Adjacent sequences: A122084 A122085 A122086 * A122088 A122089 A122090

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 19 2006

Status

approved