%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