A127742 Triangle read by rows with shape A000041 which refines the Catalan triangle A033184 using sequence A048996.

1, 1, 1, 2, 2, 1, 5, 4, 1, 3, 1, 14, 10, 4, 6, 3, 4, 1, 42, 28, 10, 4, 15, 12, 1, 8, 6, 5, 1, 132, 84, 28, 20, 42, 30, 12, 6, 20, 24, 4, 10, 10, 6, 1, 429, 264, 84, 56, 25, 126, 84, 60, 15, 12, 56, 60, 24, 24, 1, 25, 40, 10, 12, 15, 7, 1, 1430, 858, 264, 168, 140, 396, 252, 168, 75

Offset: 1

Author

Alford Arnold, Feb 23 2007

Keywords

• nonn
• tabf

Comments

Contribution from R. J. Mathar, Jul 16 2010: (Start)

The entries count Dyck paths of length 2n which have a step/stride pattern between consecutive returns to the horizontal axes (after sorting) equivalent to the k-th partition of n.

Equivalent means that each distance between two x (where y=0) is divided by 2 prior to comparison.

Example: if the x-values are (0,4,8,10,16) with 2n=16, the strides are 4,4,2,6, equal to 2,2,1,3 after division by 2, and contribute to the 1^1,2^2,3^1 partition T(8,14). (End)

Examples

			The triangle begins
1
1 1
2 2 1
5 4 1 3 1
14 10 4 6 3 4 1
etc
		

Extensions

More terms from R. J. Mathar, Jul 16 2010