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

Author

Seiichi Manyama, Apr 30 2025

Keywords

• nonn
• tabl

Examples

			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, ...
		

Crossrefs

Main diagonal gives A376791.

Programs

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);

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).