cp's OEIS Frontend

A143349 Triangle read by rows: A000012 * A054524 = A000012 * A051731 * A128407.

%I A143349 #14 Jun 05 2023 07:21:01
%S A143349 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 A143349 -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 A143349 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 A143349 Triangle read by rows: A000012 * A054524 = A000012 * A051731 * A128407.
%C A143349 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 A143349     2;
%C A143349     4, -1;
%C A143349     7, -1, -1;
%C A143349    10, -2, -1,  0;
%C A143349    ...
%C A143349 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 A143349 First few rows of the triangle:
%e A143349    1;
%e A143349    2, -1;
%e A143349    3, -1, -1;
%e A143349    4, -2, -1,  0;
%e A143349    5, -2, -1,  0, -1;
%e A143349    6, -3, -2,  0, -1,  1;
%e A143349    7, -3, -2,  0, -1,  1, -1;
%e A143349    8, -4, -2,  0, -1,  1, -1,  0;
%e A143349    9, -4, -3,  0, -1,  1, -1,  0,  0;
%e A143349   10, -5, -3,  0, -2,  1, -1,  0,  0,  1;
%e A143349   11, -5, -3,  0, -2,  1, -1,  0,  0,  1, -1;
%e A143349   12, -6, -4,  0, -2,  2, -1,  0,  0,  1, -1,  0;
%e A143349   13, -6, -4,  0, -2,  2, -1,  0,  0,  1, -1,  0, -1;
%e A143349   14, -7, -4,  0, -2,  2, -2,  0,  0,  1, -1,  0, -1,  1;
%e A143349   ...
%Y A143349 Cf. A000012, A051731, A054524, A128407, A143350.
%K A143349 tabl,sign
%O A143349 1,2
%A A143349 _Gary W. Adamson_, Aug 10 2008
%E A143349 a(39) ff. corrected by _Georg Fischer_, Jun 05 2023