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

0,4

Links

Table of n, a(n) for n=0..65.

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).

Example

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, ...

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

Crossrefs

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

Main diagonal gives A122680.

Cf. A007318, A036355, A383477.

Sequence in context: A308771 A007495 A122385 * A035002 A032578 A378905

Adjacent sequences: A383471 A383472 A383473 * A383475 A383476 A383477

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 27 2025

Status

approved