A122085 Triangle read by rows: T(n,k) = number of unlabeled free bicolored trees with n nodes (n >= 1) and k (1 <= k <= n-1, except k=0 or 1 if n=1, k=1 if n=2) nodes of one color and n-k nodes of the other color (the colors are not interchangeable).

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 7, 7, 3, 1, 1, 3, 10, 14, 10, 3, 1, 1, 4, 14, 28, 28, 14, 4, 1, 1, 4, 19, 45, 65, 45, 19, 4, 1, 1, 5, 24, 73, 132, 132, 73, 24, 5, 1, 1, 5, 30, 105, 242, 316, 242, 105, 30, 5, 1, 1, 6, 37, 152, 412, 693, 693, 412, 152

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
1 0 1
Total( 1) = 2
1 1 1
Total( 2) = 1
1 2 1
2 1 1
Total( 3) = 2
1 3 1
2 2 1
3 1 1
Total( 4) = 3
1 4 1
2 3 2
3 2 2
4 1 1
Total( 5) = 6
1 5 1
2 4 2
3 3 4
4 2 2
5 1 1
Total( 6) = 10
.
From _Andrew Howroyd_, Nov 02 2019: (Start)
Triangle for n >= 2, 1 <= k < n:
   2 | 1;
   3 | 1, 1;
   4 | 1, 1,  1;
   5 | 1, 2,  2,   1;
   6 | 1, 2,  4,   2,   1;
   7 | 1, 3,  7,   7,   3,   1;
   8 | 1, 3, 10,  14,  10,   3,   1;
   9 | 1, 4, 14,  28,  28,  14,   4,   1;
  10 | 1, 4, 19,  45,  65,  45,  19,   4,  1;
  11 | 1, 5, 24,  73, 132, 132,  73,  24,  5, 1;
  12 | 1, 5, 30, 105, 242, 316, 242, 105, 30, 5, 1;
  ...
(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 A122086.

Cf. A329054 (regular array with same data).