A127742 - OEIS

%I #7 Jan 14 2013 21:48:21

%S 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,

%T 84,28,20,42,30,12,6,20,24,4,10,10,6,1,429,264,84,56,25,126,84,60,15,

%U 12,56,60,24,24,1,25,40,10,12,15,7,1,1430,858,264,168,140,396,252,168,75

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

%C Contribution from _R. J. Mathar_, Jul 16 2010: (Start)

%C 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.

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

%C 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)

%e The triangle begins

%e 1

%e 1 1

%e 2 2 1

%e 5 4 1 3 1

%e 14 10 4 6 3 4 1

%e etc

%K nonn,tabf

%O 1,4

%A _Alford Arnold_, Feb 23 2007

%E More terms from _R. J. Mathar_, Jul 16 2010