A133807 Triangle read by rows: A007318 * (A097806 + A133566 - I) as infinite lower triangular matrices, where I is the identity matrix.

1, 2, 1, 3, 4, 1, 4, 9, 4, 1, 5, 16, 10, 6, 1, 6, 25, 20, 20, 6, 1, 7, 36, 35, 50, 21, 8, 1, 8, 49, 56, 105, 56, 35, 8, 1, 9, 64, 84, 196, 126, 112, 36, 10, 1, 10, 81, 120, 336, 252, 294, 120, 54, 10, 1, 11, 100, 165, 540, 462, 672, 330, 210, 55, 12, 1

Offset

0,2

Comments

The matrix M = (A097806 + A133566 - I) is the triangle with (1,1,1,...) in the main diagonal, (1,2,1,2,1,...) in the subdiagonal and the rest zeros.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Formula

From Andrew Howroyd, Sep 24 2025: (Start)

T(n,k) = binomial(n,k) + (3*binomial(n,k+1) - (-1)^k*binomial(n,k+1))/2.

G.f.: (1 - (1 - y)*x + y*x^2)/((1 - x)*(1 - (1 - y)*x)*(1 - (1 + y)*x)). (End)

Example

First few rows of the triangle:
  1;
  2,  1;
  3,  4,  1;
  4,  9,  4,  1;
  5, 16, 10,  6,  1;
  6, 25, 20, 20,  6,  1;
  7, 36, 35, 50, 21,  8,  1;
  ...

Prog

(PARI) T(n, k) = binomial(n, k) + (3*binomial(n, k+1) - (-1)^k*binomial(n, k+1))/2 \\ Andrew Howroyd, Sep 25 2025

Crossrefs

Row sums are A131051.

Cf. A007318, A097806, A133566, A133601.

Sequence in context: A297224 A180383 A374896 * A325001 A093375 A103283

Adjacent sequences: A133804 A133805 A133806 * A133808 A133809 A133810

Keyword

nonn,tabl

Author

Gary W. Adamson, Sep 23 2007

Extensions

Offset changed and a(55) onwards from Andrew Howroyd, Sep 24 2025

Status

approved