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Triangle read by rows, A027293 * an infinite lower triangular matrix with A147843 as the diagonal and the rest zeros.
2

%I #33 Jan 12 2023 01:34:22

%S 1,1,2,2,2,0,3,4,0,0,5,6,0,0,-5,7,10,0,0,-5,0,11,14,0,0,-10,0,-7,15,

%T 22,0,0,-15,0,-7,0,22,30,0,0,-25,0,-14,0,0,30,44,0,0,-35,0,-21,0,0,0,

%U 42,60,0,0,-55,0,-35,0,0,0,0,56,84,0,0,-75,0,-49,0,0,0,0,12

%N Triangle read by rows, A027293 * an infinite lower triangular matrix with A147843 as the diagonal and the rest zeros.

%C Left border = the partition numbers, A000041; right border = A147843 starting (1, 2, 0, ...).

%C Row sums apparently give A000203. Check: Sum of row 6 terms = [7, 5, 3, 2, 1, 1] dot [1, 2, 0, 0, -5, 0] = [7 + 10 + 0 + 0 -5 + 0] = 12 = A000203(6).

%F Equals triangle A027293 * a lower triangular matrix with A147843 (deleting the first zero) as the right border and the rest zeros.

%F T(n,k) = A147843(k) * A027293(n,k). - _Joerg Arndt_, Dec 29 2022

%e First few rows of the triangle:

%e 1;

%e 1, 2;

%e 2, 2, 0;

%e 3, 4, 0, 0;

%e 5, 6, 0, 0, -5;

%e 7, 10, 0, 0, -5, 0;

%e 11, 14, 0, 0, -10, 0, -7;

%e 15, 22, 0, 0, -15, 0, -7, 0;

%e 22, 30, 0, 0, -25, 0, -14, 0, 0;

%e 30, 44, 0, 0, -35, 0, -21, 0, 0, 0;

%e 42, 60, 0, 0, -55, 0, -35, 0, 0, 0, 0;

%e 56, 84, 6, 0, -75, 0, -49, 0, 0, 0, 0, 12;

%e ...

%Y Cf. A027293, A147843, A000041, A000203.

%K tabl,sign

%O 1,3

%A _Gary W. Adamson_, Mar 28 2010

%E Terms corrected by _Gary W. Adamson_, Dec 27 2022