A383551 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0),(0,1),(3,3).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 21, 15, 6, 1, 1, 7, 21, 37, 37, 21, 7, 1, 1, 8, 28, 59, 76, 59, 28, 8, 1, 1, 9, 36, 88, 138, 138, 88, 36, 9, 1, 1, 10, 45, 125, 230, 282, 230, 125, 45, 10, 1, 1, 11, 55, 171, 360, 522, 522, 360, 171, 55, 11, 1

Offset

0,5

Links

Table of n, a(n) for n=0..77.

Formula

A(n,k) = A(k,n).

A(n,k) = A(n-1,k) + A(n,k-1) + A(n-3,k-3).

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

Example

Square array A(n,k) begins:
  1, 1,  1,  1,   1,   1,    1, ...
  1, 2,  3,  4,   5,   6,    7, ...
  1, 3,  6, 10,  15,  21,   28, ...
  1, 4, 10, 21,  37,  59,   88, ...
  1, 5, 15, 37,  76, 138,  230, ...
  1, 6, 21, 59, 138, 282,  522, ...
  1, 7, 28, 88, 230, 522, 1065, ...

Prog

(PARI) a(n, k) = my(x='x+O('x^(n+1)), y='y+O('y^(k+1))); polcoef(polcoef(1/(1-x-y-x^3*y^3), n), k);

Crossrefs

Main diagonal gives A376791.

Sequence in context: A130595 A383566 A108363 * A329052 A076831 A197061

Adjacent sequences: A383548 A383549 A383550 * A383552 A383553 A383554

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 30 2025

Status

approved