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Triangle read by rows: A000012^(-2) * A027293 as infinite lower triangular matrices.
1

%I #14 Jul 29 2023 03:28:38

%S 1,-1,1,1,-1,1,0,1,-1,1,1,0,1,-1,1,0,1,0,1,-1,1,2,0,1,0,1,-1,1,0,2,0,

%T 1,0,1,-1,1,3,0,2,0,1,0,1,-1,1,1,3,0,2,0,1,0,1,-1,1,4,1,3,0,2,0,1,0,1,

%U -1,1,2,4,1,3,0,2,0,1,0,1,-1,1

%N Triangle read by rows: A000012^(-2) * A027293 as infinite lower triangular matrices.

%C Row sums = A002865, (1, 0, 1, 1, 2, 2, 4, 4, 7, 8,...).

%C This triangle is the lower right half of a Toeplitz matrix. Each column of this triangle has the form [1, -1] U A053445. - _Georg Fischer_, Jul 28 2023

%F A000012^(-1) is the pairwise difference operator, and A027293 = a triangle with A000041 in every column.

%F Equals A185018 * A027293 since A000012^2 = A004736 and A004736^(-1) = A185018. - _Georg Fischer_, Jul 28 2023

%e First few rows of the triangle =

%e 1;

%e -1, 1;

%e 1, -1, 1;

%e 0, 1, -1, 1;

%e 1, 0, 1, -1, 1;

%e 0, 1, 0, 1, -1, 1;

%e 2, 0, 1, 0, 1, -1, 1;

%e 0, 2, 0, 1, 0, 1, -1, 1;

%e 3, 0, 2, 0, 1, 0, 1, -1, 1;

%e 1, 3, 0, 2, 0, 1, 0, 1, -1, 1;

%e 4, 1, 3, 0, 2, 0, 1, 0, 1, -1, 1;

%e 2, 4, 1, 3, 0, 2, 0, 1, 0, 1, -1, 1;

%e ...

%Y Cf. A000012, A000041, A002865, A004736, A027293, A053445, A185018.

%K tabl,sign

%O 0,22

%A _Gary W. Adamson_, Nov 11 2008