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%I #14 Jun 05 2023 07:21:01
%S 1,2,-1,3,-1,-1,4,-2,-1,0,5,-2,-1,0,-1,6,-3,-2,0,-1,1,7,-3,-2,0,-1,1,
%T -1,8,-4,-2,0,-1,1,-1,0,9,-4,-3,0,-1,1,-1,0,0,10,-5,-3,0,-2,1,-1,0,0,
%U 1,11,-5,-3,0,-2,1,-1,0,0,1,-1,12,-6,-4,0,-2,2,-1,0,0,1,-1,0,13,-6,-4,0,-2,2,-1,0,0,1,-1,0,-1,14,-7,-4,0,-2,2,-2,0,0,1,-1,0,-1,1
%N Triangle read by rows: A000012 * A054524 = A000012 * A051731 * A128407.
%C The triangle acts as a transform converting any sequence S(k) into a triangle with row sums = S(k). By way of example, begin with S(k), the primes: (2, 3, 5, 7, 11, ...). Add (0, 1, 2, 3, 4, ...) to the sequence getting (prime(n)+(n-1)) = (2, 4, 7, 10, 15, 18, 23, 36, 31, ...) = sequence Q(k). Then replace column 1 (1, 2, 3, ...) of triangle A143349 with sequence Q(k). This = triangle A143350 with row sums prime(n):
%C 2;
%C 4, -1;
%C 7, -1, -1;
%C 10, -2, -1, 0;
%C ...
%C The A000012 multiplier takes partial sums of A054524 column terms. A051731 is the inverse Mobius transform and A128407 = an infinite lower triangular matrix with mu(n) in the main diagonal and the rest zeros.
%e First few rows of the triangle:
%e 1;
%e 2, -1;
%e 3, -1, -1;
%e 4, -2, -1, 0;
%e 5, -2, -1, 0, -1;
%e 6, -3, -2, 0, -1, 1;
%e 7, -3, -2, 0, -1, 1, -1;
%e 8, -4, -2, 0, -1, 1, -1, 0;
%e 9, -4, -3, 0, -1, 1, -1, 0, 0;
%e 10, -5, -3, 0, -2, 1, -1, 0, 0, 1;
%e 11, -5, -3, 0, -2, 1, -1, 0, 0, 1, -1;
%e 12, -6, -4, 0, -2, 2, -1, 0, 0, 1, -1, 0;
%e 13, -6, -4, 0, -2, 2, -1, 0, 0, 1, -1, 0, -1;
%e 14, -7, -4, 0, -2, 2, -2, 0, 0, 1, -1, 0, -1, 1;
%e ...
%Y Cf. A000012, A051731, A054524, A128407, A143350.
%K tabl,sign
%O 1,2
%A _Gary W. Adamson_, Aug 10 2008
%E a(39) ff. corrected by _Georg Fischer_, Jun 05 2023
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