A383550 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 (2,0),(0,2),(3,3).

1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 3, 1, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 4, 2, 6, 2, 4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 5, 3, 10, 6, 10, 3, 5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 4, 15, 12, 21, 12, 15, 4, 6, 0, 1

Offset

0,13

Links

Table of n, a(n) for n=0..90.

Formula

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

If n - k == 1 (mod 2), A(n,k) = 0.

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

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

Example

Square array A(n,k) begins:
  1, 0, 1, 0,  1,  0,  1,  0,  1, ...
  0, 0, 0, 0,  0,  0,  0,  0,  0, ...
  1, 0, 2, 0,  3,  0,  4,  0,  5, ...
  0, 0, 0, 1,  0,  2,  0,  3,  0, ...
  1, 0, 3, 0,  6,  0, 10,  0, 15, ...
  0, 0, 0, 2,  0,  6,  0, 12,  0, ...
  1, 0, 4, 0, 10,  0, 21,  0, 38, ...
  0, 0, 0, 3,  0, 12,  0, 30,  0, ...
  1, 0, 5, 0, 15,  0, 38,  0, 82, ...

Prog

(PARI) a(n, k) = my(x='x+O('x^(n+1)), y='y+O('y^(k+1))); polcoef(polcoef(1/(1-x^2-y^2-x^3*y^3), n), k);

Crossrefs

Main diagonal gives A053442.

Cf. A027907, A383567.

Sequence in context: A325045 A204293 A383567 * A206479 A219484 A060396

Adjacent sequences: A383547 A383548 A383549 * A383551 A383552 A383553

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 30 2025

Status

approved