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

Author

N. J. A. Sloane, Jun 15 2002

Keywords

• nonn
• easy
• tabl

Examples

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

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.

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.

Programs

Maple

T:= proc(n, k) option remember; `if`(n=0 and k=0, 1,
     `if`(k<0 or nAlois 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 *)

Extensions

More terms from Joshua Zucker, May 10 2006