A131048 Triangle read by rows: (1/3) * (A007318^2 - A007318^(-1)) as infinite lower triangular matrices.

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, 85, 344, 588, 616, 350, 168, 28, 8, 171, 765, 1548, 1764, 1386, 630, 252, 36, 9, 341, 1710, 3825, 5160, 4410, 2772, 1050, 360, 45, 10, 683, 3751, 9405, 14025, 14190, 9702, 5082, 1650, 495, 55, 11

Offset

1,3

Comments

Let P = Pascal's triangle, A007318. Then form (1/3) * (P^2 - 1/P). Exclude the initial row of zeros. The resulting triangle T(n,k) is for n >= 1, 0 <= k < n.

Analogous triangles for other powers of P are: A131047, A131049 and A131050.

Row sums are powers of 3.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Formula

From Peter Bala, Oct 24 2007: (Start)

O.g.f.: 1/(1 - (2*x + 1)*t + (x^2 + x - 2)*t^2) = 1 + (1 + 2*x)*t + (3 + 3*x + 3*x^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. (End)

Example

First few rows of the triangle:
   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;
  ...

Prog

(PARI) M(n)={my(t=matrix(n+1, n+1, i, j, binomial(i-1, j-1))); (t^2 - t^(-1))[2..n+1, 1..n]/3}
{ my(A=M(10)); for(i=1, #A, print(A[i, 1..i])) } \\ Andrew Howroyd, Sep 23 2025
(PARI) T(n, k)=binomial(n, k)*(2^(n-k) - (-1)^(n-k)) / 3
{ for(n=1, 10, print(vector(n, k, T(n, k-1)))) } \\ Andrew Howroyd, Sep 23 2025

Crossrefs

Column 0 is A001045.

Row sums are A000244(n-1).

Cf. A007318, A131047, A131049, A131050.

Cf. A094440, A132148.

Sequence in context: A343885 A264870 A346048 * A119688 A126868 A373093

Adjacent sequences: A131045 A131046 A131047 * A131049 A131050 A131051

Keyword

nonn,tabl

Author

Gary W. Adamson, Jun 12 2007

Status

approved