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

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, -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, 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

Offset: 1

Author

Gary W. Adamson, Aug 10 2008

Keywords

• tabl
• sign

Comments

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):

2;

4, -1;

7, -1, -1;

10, -2, -1, 0;

...

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.

Examples

			First few rows of the triangle:
   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, -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,  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;
  ...
		

Crossrefs

Cf. A000012, A051731, A054524, A128407, A143350.

Extensions

a(39) ff. corrected by Georg Fischer, Jun 05 2023