A383477 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).

1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 9, 12, 7, 1, 5, 14, 25, 26, 13, 1, 6, 20, 44, 63, 56, 24, 1, 7, 27, 70, 125, 153, 118, 44, 1, 8, 35, 104, 220, 336, 359, 244, 81, 1, 9, 44, 147, 357, 646, 864, 819, 499, 149, 1, 10, 54, 200, 546, 1134, 1800, 2144, 1830, 1010, 274

Offset

0,5

Links

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

Formula

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

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

G.f. of column k: 1 / (1 - x - x^2 - x^3)^(k+1).

Example

Square array A(n,k) begins:
   1,   1,   1,   1,    1,    1,    1, ...
   1,   2,   3,   4,    5,    6,    7, ...
   2,   5,   9,  14,   20,   27,   35, ...
   4,  12,  25,  44,   70,  104,  147, ...
   7,  26,  63, 125,  220,  357,  546, ...
  13,  56, 153, 336,  646, 1134, 1862, ...
  24, 118, 359, 864, 1800, 3395, 5950, ...

Prog

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

Crossrefs

Column k=0..2 give A000073(n+2), A073778(n+4), A292326(n-1).

Main diagonal gives A383478.

Cf. A038137, A383474.

Sequence in context: A079956 A140717 A257005 * A160232 A026300 A099514

Adjacent sequences: A383474 A383475 A383476 * A383478 A383479 A383480

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 28 2025

Status

approved