login
Triangle read by rows, generated from the Narayana triangle as a matrix.
2

%I #25 Sep 02 2024 00:20:06

%S 1,1,1,1,2,1,1,4,2,1,1,7,5,2,1,1,11,13,5,2,1,1,16,31,14,5,2,1,1,22,66,

%T 41,14,5,2,1,1,29,127,116,42,14,5,2,1,1,37,225,302,131,42,14,5,2,1,1,

%U 46,373,715,407,132,42,14,5,2,1,1,56,586,1549,1205,428,132,42,14,5,2,1

%N Triangle read by rows, generated from the Narayana triangle as a matrix.

%C The n-th column starting from the top has the first n terms in the Catalan sequence: (1, 2, 5, 14, ...)

%F n-th column (offset) is generated by P * V; P = the Narayana triangle as an infinite lower triangular matrix, V = vector for n-th column comprising n leading 1's and the rest zeros (e.g., V for 3rd column = [1, 1, 1, 0, 0, 0, ...]).

%e Col. 2 offset = 1, 2, 4, 7, 11, 16, 22, ... since P * [1, 1, 0, 0, 0, ...] = 1, 2, 4, 7, 11, ...

%e First few rows of the triangle are:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 4, 2, 1;

%e 1, 7, 5, 2, 1;

%e 1, 11, 13, 5, 2, 1;

%e 1, 16, 31, 14, 5, 2, 1;

%e ...

%e Second column = A000124.

%Y Cf. A000108, A000124, A001263, A137940.

%K nonn,tabl

%O 1,5

%A _Gary W. Adamson_, May 01 2005

%E More terms from _Alois P. Heinz_, Nov 28 2021