login
Triangle by rows, generated from a triangle with (1,2,1,1,1,...) in every column.
3

%I #5 Jan 28 2022 07:47:23

%S 1,2,1,1,2,3,1,1,6,6,1,1,3,12,14,1,1,3,6,28,31,1,1,3,6,14,62,70,1,1,3,

%T 6,14,31,140,157,1,1,3,6,14,31,70,314,353,1,1,3,6,14,31,70,157,706,

%U 793,1,1,3,6,14,31,70,157,353,1586,1782

%N Triangle by rows, generated from a triangle with (1,2,1,1,1,...) in every column.

%C Row sums = A006356: (1, 3, 6, 14, 31, 70, 157, 353,...).

%C Sum of n-th row terms = rightmost term of next row.

%F Let M be an infinite Toeplitz lower triangular matrix with (1,2,1,1,1,..) in every column. A180262 = M * a diagonalized variant of A006356 such that the main diagonal = A006356 prefaced with a 1: (1, 1, 3, 6, 14, 31,...) and the rest zeros.

%e First few rows of the triangle:

%e 1;

%e 2, 1;

%e 1, 2, 3;

%e 1, 1, 6, 6;

%e 1, 1, 3, 12, 14;

%e 1, 1, 3, 6, 28, 31;

%e 1, 1, 3, 6, 14, 62, 70;

%e 1, 1, 3, 6, 14, 31, 140, 157;

%e 1, 1, 3, 6, 14, 31, 70, 314, 353;

%e 1, 1, 3, 6, 14, 31, 70, 157, 706, 793;

%e 1, 1, 3, 6, 14, 31, 70, 157, 353, 1586, 1782;

%e 1, 1, 3, 6, 14, 31, 70, 157, 353, 793, 3564, 4004;

%e 1, 1, 3, 6, 14, 31, 70, 157, 353, 793, 1782, 8008, 8997;

%e 1, 1, 3, 6, 14, 31, 70, 157, 353, 793, 1782, 4004, 17994, 20216;

%e ...

%e Example: row 3 of the triangle = (1, 1, 6, 6) = termwise products of (1, 1, 2, 1) and (1, 1, 3, 6).

%Y Cf. A006356.

%K nonn,tabl

%O 0,2

%A _Gary W. Adamson_, Aug 21 2010