login
Triangle read by rows, generated from A001263.
1

%I #13 May 04 2024 09:24:05

%S 1,1,1,1,2,1,1,3,5,1,1,4,12,14,1,1,5,22,57,42,1,1,6,35,148,303,132,1,

%T 1,7,51,305,1144,1743,429,1,8,70,546,3105,9784,10629,1430,1

%N Triangle read by rows, generated from A001263.

%C Rows of the array are row sums of n-th powers of the Narayana triangle; e.g., row 1 = A000108: (1, 2, 5, 14, 42, ...); row 2 = row sums of the Narayana triangle squared (A103370): (1, 3, 12, 57, 303, ...), etc.

%F Let M be the infinite lower triangular Narayana triangle (A001263). Perform M^n * [1 0 0 0 ...] getting an array. Take antidiagonals of the array which become rows of the triangle A112338.

%e In the array, antidiagonal terms (1, 3, 5, 1) become row 3 of the triangle.

%e First few rows of the array:

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 5, 14, 42, 132, ...

%e 1, 3, 12, 57, 303, 1743, ...

%e 1, 4, 22, 148, 1144, 9784, ...

%e 1, 5, 35, 305, 3105, 35505, ...

%e First few rows of the triangle:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 3, 5, 1;

%e 1, 4, 12, 14, 1;

%e 1, 5, 22, 57, 42, 1;

%e 1, 6, 35, 148, 303, 132, 1;

%Y Cf. A001263, A000326, A005915, A095266, A000108, A103370.

%K nonn,tabl

%O 0,5

%A _Gary W. Adamson_, Sep 04 2005