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%I #4 Mar 10 2015 02:11:38
%S 1,2,-1,3,0,-1,4,-2,0,0,5,0,0,0,-1,6,-3,-2,0,0,1,7,0,0,0,0,0,-1,8,-4,
%T 0,0,0,0,0,0,9,0,-3,0,0,0,0,0,0,10,-5,0,0,-2,0,0,0,0,1,11,0,0,0,0,0,0,
%U 0,0,0,-1,12,-6,-4,0,0,2,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,-1,14,-7,0,0,0,0,-2,0,0,0,0,0,0,1
%N Triangle read by rows, A126988 * A128407 as infinite lower triangular matrices.
%C Row sums = A000010, phi(n): (1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4,...); as a consequence of the Dedekind-Liouville rule illustrated in the example and on p. 137 of "Concrete Mathematics".
%D Ronald L. Graham, Donald E. Knuth & Oren Patashnik, "Concrete Mathematics" 2nd ed.; Addison-Wesley, 1994, p. 137.
%F Triangle read by rows generated from the Dedekind-Liouville rule: T(n,k) = mu(k)*(n/k) if k divides n. T(n,k) = 0 if k is not a divisor of n. Equals A126988 * A128407
%e First few rows of the triangle are:
%e 1;
%e 2, -1;
%e 3, 0, -1;
%e 4, -2, 0, 0;
%e 5, 0, 0, 0, -1;
%e 6, -3, -2, 0, 0, 1;
%e 7, 0, 0, 0, 0, 0, -1;
%e 8, -4, 0, 0, 0, 0, 0, 0;
%e 9, 0, -3, 0, 0, 0, 0, 0, 0;
%e 10, -5, 0, 0, -2, 0, 0, 0, 0, 1;
%e 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1;
%e 12, -6, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0;
%e 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1;
%e 14, -7, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1;
%e ...
%e Row 12 = (12, -6, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0) since (Cf. A126988 - the divisors of 12 are (12, 6, 4, 3, 0, 2, 0, 0, 0, 0, 0, 1) and applying mu(k) * (nonzero terms), we get (1*12, (-1)*6, (-1)*4, 1*2) sum = 4 = phi(12).
%Y Cf. A000010, A128407, A126988, A008683.
%K tabl,sign
%O 1,2
%A _Gary W. Adamson_, Aug 01 2008
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