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

1,10

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

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
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)

Crossrefs

Row sums give A122086.

Cf. A329054 (regular array with same data).

Sequence in context: A172453 A172479 A339788 * A209612 A209805 A238453

Adjacent sequences: A122082 A122083 A122084 * A122086 A122087 A122088

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 19 2006

Status

approved