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Triangle read by rows, generated from (1, 2, 3, ...).
1

%I #13 Feb 07 2022 03:42:52

%S 1,1,3,1,4,6,1,5,11,10,1,6,18,26,15,1,7,27,58,57,21,1,8,38,112,179,

%T 120,28,1,9,51,194,453,543,247,36

%N Triangle read by rows, generated from (1, 2, 3, ...).

%C By diagonals (d=1,2,3,...) going to the left with (1,3,6,...) = d(1), these are sequences of the form (k-th term a(k) = d*a(k-1) + k). Example: 1, 7, 38, 194, ... (the 5th diagonal) = A014827, is generated by a(k) = 5*a(k-1) + k. Diagonal 2 = (1, 4, 11, 26, ...) = A000295; Diagonal 3 = (1, 5, 18, ...) = A000340; Diagonal 4 = (1, 6, 27, ...) = A014825.

%C Triangle A108243 is generated by analogous operations from (..., 3, 2, 1) instead of (1, 2, 3, ...).

%F n-th column = f(x), x = 1, 2, 3, ...; x^(n) + 2*x^(n-1) + 3*x^(n-2) + ... + (n+1).

%e 4th column (offset) = 10, 26, 58, 112, ...= f(x), x = 1, 2, 3; x^3 + 2x^2 + 3x + 4.

%e First few rows of the triangle are:

%e 1;

%e 1, 3;

%e 1, 4, 6;

%e 1, 5, 11, 10;

%e 1, 6, 18, 26, 15;

%e 1, 7, 27, 58, 57, 21;

%e 1, 8, 38, 112, 179, 120, 28;

%e ...

%Y Cf. A108286, A000295, A000349, A014825, A000217, A108283, A108284, A108286.

%K nonn,tabl

%O 0,3

%A _Gary W. Adamson_, May 30 2005