1,3

Left border = A001045: (1, 1, 3, 5, 11, 21, 43, 85,...). Row sums = (1, 3, 9, 27,...). Analogous triangles for other powers of P are: A131047, A131049, A131050 and A131051.

Let A007318 (Pascal's triangle) = P. then A131048 = (1/3) * (P^2 - 1/P). Delete right border of zeros.

O.g.f.: 1/(1-(2x+1)*t+(x^2+x-2)*t^2) = 1+(1+2x)*t+(3+3x+3x^2)*t^2+ ... . T(n,n-k) = (1/3)*C(n,k)*(2^k - (-1)^k) = C(n,k)*A001045(k). The row polynomials R(n,x):= sum {k = 0..n} T(n,n-k)*x^(n-k) = (1/3)*((x+2)^n-(x-1)^n) and have the divisibility property R(n,x) divides R(m,x) in the polynomial ring Z[x] if n divides m. The polynomials R(n,-x), n >= 2, satisfy a Riemann hypothesis: their zeros lie on the vertical line Re x = 1/2 in the complex plane. Compare with A094440. - Peter Bala (pbala(AT)toucansurf.com), Oct 24 2007

First few rows of the triangle are:

1;

1, 2;

3, 3, 3;

5, 12, 6, 4;

11, 25, 30, 10, 5;

21, 66, 75, 60, 15, 6;

43, 147, 231, 175, 105, 21, 7;

...

Cf. A131047, A131049, A131050, A131051, A001045, A007318.

Cf. A001045, A094440, A132148.

Sequence in context: A036029 A035362 A042957 this_sequence A126868 A119688 A134187

Adjacent sequences: A131045 A131046 A131047 this_sequence A131049 A131050 A131051

nonn,tabl

Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 12 2007