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

1,4

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)

Links

Table of n, a(n) for n=1..75.

Example

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

Crossrefs

Sequence in context: A324798 A226059 A353738 * A110438 A184051 A121460

Adjacent sequences: A127739 A127740 A127741 * A127743 A127744 A127745

Keyword

nonn,tabf

Author

Alford Arnold, Feb 23 2007

Extensions

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

Status

approved