A071946 Triangle T(n,k) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using only steps R = (1,0), V = (0,1) and D = (3,1).

1, 1, 1, 1, 2, 2, 1, 4, 6, 6, 1, 6, 13, 19, 19, 1, 8, 23, 44, 63, 63, 1, 10, 37, 87, 156, 219, 219, 1, 12, 55, 155, 330, 568, 787, 787, 1, 14, 77, 255, 629, 1260, 2110, 2897, 2897, 1, 16, 103, 395, 1111, 2527, 4856, 7972, 10869, 10869, 1, 18, 133, 583, 1849, 4706, 10130, 18889, 30545, 41414, 41414

Offset

0,5

Links

Alois P. Heinz, Rows n = 0..150, flattened

D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.

Formula

T(n,k) = T(n,k-1) + T(n-1,k) + T(n-3,k-1), with T(0,0) = 1, T(n,k) = T(n,n-k) = 0 for k<0. - Leonidas Liponis, Dec 20 2025

Example

Triangle T(n,k) begins:
  1;
  1,  1;
  1,  2,  2;
  1,  4,  6,  6;
  1,  6, 13, 19, 19;
  1,  8, 23, 44, 63, 63;
  ...

Maple

T:= proc(n, k) option remember; `if`(n=0 and k=0, 1,
     `if`(k<0 or n<k, 0, T(n-1, k)+T(n, k-1)+T(n-3, k-1)))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, May 05 2023

Mathematica

T[n_, k_] := T[n, k] = If[n == 0 && k == 0, 1,
   If[k < 0 || n < k, 0, T[n-1, k] + T[n, k-1] + T[n-3, k-1]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 25 2025, after Alois P. Heinz *)

Crossrefs

Related arrays: A071943, A071944, A071945.

A108076 is the reverse, A119254 is the row sums and A071969 is the last (largest) number in each row.

Sequence in context: A191490 A061598 A328873 * A053495 A096747 A299504

Adjacent sequences: A071943 A071944 A071945 * A071947 A071948 A071949

Keyword

nonn,easy,tabl,changed

Author

N. J. A. Sloane, Jun 15 2002

Extensions

More terms from Joshua Zucker, May 10 2006

Status

approved