A383448 Irregular triangle read by rows: T(n,k) (n>=1, k>=0) is the number of trees with n nodes in which there are k edges whose end-vertices have the same degree.

1, 0, 1, 1, 1, 1, 2, 0, 1, 3, 2, 0, 1, 6, 3, 1, 0, 1, 9, 7, 5, 1, 0, 1, 19, 12, 10, 4, 1, 0, 1, 33, 33, 18, 15, 5, 1, 0, 1, 67, 66, 54, 26, 15, 5, 1, 0, 1, 130, 154, 128, 77, 36, 18, 6, 1, 0, 1, 270, 344, 309, 199, 110, 40, 21, 6, 1, 0, 1, 547, 806, 752, 530, 294, 147, 50, 24, 7, 1, 0, 1

Offset: 1

Author

N. J. A. Sloane, May 01 2025

Keywords

• nonn
• tabf

Comments

T(n,0) is the number of "peerless" trees on n nodes (see A383447). The row sums are A000055.

For n >= 3, row n has n-2 entries.

Examples

			Triangle begins:
   1,
   0, 1,
   1,
   1, 1,
   2, 0, 1,
   3, 2, 0, 1,
   6, 3, 1, 0, 1,
   9, 7, 5, 1, 0, 1,
   19, 12, 10, 4, 1, 0, 1,
   33, 33, 18, 15, 5, 1, 0, 1,
   67, 66, 54, 26, 15, 5, 1, 0, 1,
   130, 154, 128, 77, 36, 18, 6, 1, 0, 1,
   270, 344, 309, 199, 110, 40, 21, 6, 1, 0, 1,
   547, 806, 752, 530, 294, 147, 50, 24, 7, 1, 0, 1
   ...
Enough rows are shown to demonstrate that the leading entry only dominates for small n. - _N. J. A. Sloane_, May 04 2025
		

References

• F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 233.

Links

• N. J. A. Sloane, Illustration for row 8 (The 23 trees with 8 nodes are numbered in black ink in the order in which they appear in Harary's table. Edges with equal degree nodes are drawn in red and their number is shown in red ink.)
• Jakub Buczak, Rows 1 to 20 of the triangle, flattened.

Crossrefs

Cf. A000055, A383447.

Extensions

More terms from Jakub Buczak, May 04 2025.