A383567 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),(5,5).

1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, 6, 0, 4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 5, 0, 10, 1, 10, 0, 5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 0, 15, 2, 20, 2, 15, 0, 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-5,k-5).

G.f.: 1 / (1 - x^2 - y^2 - x^5*y^5).

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, 0,  0, 0,  0, 0,  0, ...
  1, 0, 3, 0,  6, 0, 10, 0, 15, ...
  0, 0, 0, 0,  0, 1,  0, 2,  0, ...
  1, 0, 4, 0, 10, 0, 20, 0, 35, ...
  0, 0, 0, 0,  0, 2,  0, 6,  0, ...
  1, 0, 5, 0, 15, 0, 35, 0, 70, ...

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^5*y^5), n), k);

Crossrefs

Main diagonal gives A383568.

Cf. A027907, A383550.

Sequence in context: A118917 A325045 A204293 * A383550 A206479 A219484

Adjacent sequences: A383564 A383565 A383566 * A383568 A383569 A383570

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 30 2025

Status

approved