1,2

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 (p(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 p(n):

2;

4, -1;

7, -1, -1;

10, -2, -1, 0;

...

Triangle read by rows, A000012 * A054524 = A000012 * A051731 * A128407; 1<=k<=n. 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.

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;

...

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

Sequence in context: A113924 A084296 A062534 this_sequence A088425 A141294 A010766

Adjacent sequences: A143346 A143347 A143348 this_sequence A143350 A143351 A143352

tabl,sign

Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 10 2008