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%I #20 Jul 29 2019 12:25:36
%S 1,-1,2,-1,-2,6,0,-4,0,8,-1,-2,-3,-4,20,1,-2,-6,-4,5,12,-1,-2,-3,-4,
%T -5,-6,42,0,0,0,-16,0,0,0,32,0,0,-9,0,0,-18,0,0,54,1,-2,3,-4,-20,-6,7,
%U -8,9,40,-1,-2,-3,-4,-5,-6,-7,-8,-9,-10,110
%N Triangle read by rows, A054533 * A127648 (matrix product).
%C Row sums = A023896: (1, 1, 3, 4, 10, 16, 21, ...).
%C Right border = A002618: (1, 2, 6, 8, 20, 12, ...).
%C Left border = mu(n) = A008683 (n).
%H Jinyuan Wang, <a href="/A144824/b144824.txt">Table of n, a(n) for n = 1..10000</a>
%F Triangle read by rows, A054533 * A127648 (matrix product). The operation is equivalent to taking termwise products of row A054533 terms and the natural numbers.
%F T(n, k) = k * Sum_{d|gcd(n,k)} d * mu(n/d) for n >= 1 and 1 <= k <= n. - _Petros Hadjicostas_, Jul 28 2019
%F a(n) = A002260(n)*A054533(n). - _Jinyuan Wang_, Jul 29 2019
%e Triangle A054533 starts as follows:
%e 1;
%e -1, 1;
%e -1, -1, 2;
%e 0, -2, 0, 2;
%e -1, -1, -1, -1, 4;
%e 1, -1, -2, -1, 1, 2;
%e ...
%e The first few rows of triangle A144824 are as follows:
%e 1;
%e -1, 2;
%e -1, -2, 6;
%e 0, -4, 0, 8;
%e -1, -2, -3, -4, 20;
%e 1, -2, -6, -4, 5, 12;
%e -1, -2, -3, -4, -5, -6, 42;
%e ...
%Y Cf. A002260, A002618, A008683, A023896, A054533, A127648.
%K tabl,sign
%O 1,3
%A _Gary W. Adamson_, Sep 21 2008
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