A133805 Triangle read by rows: A007318 * A133566 * A133080 as infinite lower triangular matrices.

1, 2, 1, 4, 3, 1, 7, 6, 4, 1, 11, 10, 11, 5, 1, 16, 15, 25, 15, 6, 1, 22, 21, 50, 35, 22, 7, 1, 29, 28, 91, 70, 63, 28, 8, 1, 37, 36, 154, 126, 154, 84, 37, 9, 1, 46, 45, 246, 210, 336, 210, 129, 45, 10, 1, 56, 55, 375, 330, 672, 462, 375, 165, 56, 11, 1, 67, 66, 550, 495, 1254, 924, 957, 495, 231, 66, 12, 1

Offset

0,2

Comments

The matrix (A133566 * A133080) is an infinite lower triangular matrix with (1,1,1,...) in the main and subdiagonals and (1,0,1,0,1,...) 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 - (1 - y)*x + x^2)/((1 - x)*(1 - (1 + y)*x)*(1 - (1 - y)*x)). (End)

Example

First few rows of the triangle:
   1;
   2,  1;
   4,  3,  1;
   7,  6,  4,  1;
  11, 10, 11,  5,  1;
  16, 15, 25, 15,  6,  1;
  22, 21, 50, 35, 22,  7,  1;
  29, 28, 91, 70, 63, 28,  8,  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 A131051.

Column 0 is A000124.

Cf. A007318, A133080, A133566, A133804.

Sequence in context: A027421 A131252 A345123 * A131254 A210229 A210213

Adjacent sequences: A133802 A133803 A133804 * A133806 A133807 A133808

Keyword

nonn,tabl

Author

Gary W. Adamson, Sep 23 2007

Extensions

a(20) = 1 inserted and more terms from Georg Fischer, Jun 08 2023

Offset changed by Andrew Howroyd, Sep 25 2025

Status

approved