A133804 Triangle read by rows: A007318 * A133080 * A133566 as infinite lower triangular matrices.

1, 2, 1, 3, 3, 1, 4, 7, 4, 1, 5, 14, 10, 5, 1, 6, 25, 20, 16, 6, 1, 7, 41, 35, 41, 21, 7, 1, 8, 63, 56, 91, 56, 29, 8, 1, 9, 92, 84, 182, 126, 92, 36, 9, 1, 10, 129, 120, 336, 252, 246, 120, 46, 10, 1, 11, 175, 165, 582, 462, 582, 330, 175, 55, 11, 1, 12, 231, 220, 957, 792, 1254, 792, 550, 220, 67, 12, 1

Offset

0,2

Comments

The matrix A133080 * A133566 is an infinite lower triangular matrix with (1,1,1,...) in the main and subdiagonals and (0,1,0,1,0,...) in the subsubdiagonal.

Links

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

Formula

From Andrew Howroyd, Sep 25 2025: (Start)

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

G.f.: (1 - (2 - y)*x + (1 - y)*x^2 + y*x^3)/((1 - x)^2*(1 - (1 + y)*x)*(1 - (1 - y)*x)). (End)

Example

First few rows of the triangle:
  1;
  2,  1;
  3,  3,  1;
  4,  7,  4,  1;
  5, 14, 10,  5,  1;
  6, 25, 20, 16,  6,  1;
  7, 41, 35, 41, 21,  7,  1;
  ...

Prog

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

Crossrefs

Row sums are A133124.

Cf. A007318, A133080, A133566, A133805.

Sequence in context: A126277 A253273 A055129 * A185943 A352001 A208337

Adjacent sequences: A133801 A133802 A133803 * A133805 A133806 A133807

Keyword

nonn,tabl

Author

Gary W. Adamson, Sep 23 2007

Extensions

a(21) = 1 inserted and more terms from Georg Fischer, Jun 08 2023

Offset corrected by Andrew Howroyd, Sep 25 2025

Status

approved