A383474 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),(2,0),(3,0),(0,1),(0,2),(0,3).

1, 1, 1, 2, 2, 2, 4, 5, 5, 4, 7, 12, 14, 12, 7, 13, 26, 37, 37, 26, 13, 24, 56, 89, 106, 89, 56, 24, 44, 118, 209, 277, 277, 209, 118, 44, 81, 244, 477, 698, 784, 698, 477, 244, 81, 149, 499, 1063, 1700, 2113, 2113, 1700, 1063, 499, 149, 274, 1010, 2329, 4026, 5469, 6040, 5469, 4026, 2329, 1010, 274

Offset: 0

Author

Seiichi Manyama, Apr 27 2025

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
   1,   1,   2,    4,    7,    13,    24, ...
   1,   2,   5,   12,   26,    56,   118, ...
   2,   5,  14,   37,   89,   209,   477, ...
   4,  12,  37,  106,  277,   698,  1700, ...
   7,  26,  89,  277,  784,  2113,  5469, ...
  13,  56, 209,  698, 2113,  6040, 16497, ...
  24, 118, 477, 1700, 5469, 16497, 47332, ...
		

Crossrefs

Column k=0..1 give A000073(n+2), A073778(n+4).

Main diagonal gives A122680.

Cf. A007318, A036355, A383477.

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^2-y^2-x^3-y^3), n), k);

Formula

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

A(n,k) = A(n-1,k) + A(n-2,k) + A(n-3,k) + A(n,k-1) + A(n,k-2) + A(n,k-3).

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