A168443 Triangle, T(n,k) = number of compositions a(1),...,a(k) of n, such that a(i+1) <= a(i) + 1 for 1 <= i < k.

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 4, 4, 1, 1, 3, 6, 7, 5, 1, 1, 4, 7, 11, 11, 6, 1, 1, 4, 9, 15, 19, 16, 7, 1, 1, 5, 11, 19, 29, 31, 22, 8, 1, 1, 5, 13, 25, 39, 52, 48, 29, 9, 1, 1, 6, 15, 30, 53, 76, 88, 71, 37, 10, 1, 1, 6, 18, 37, 67, 107, 140, 142, 101, 46, 11, 1, 1, 7, 20, 44, 84, 143, 207, 245, 220, 139, 56, 12, 1

Offset: 1

Author

Vladeta Jovovic, Nov 25 2009

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
  1;
  1, 1;
  1, 2, 1;
  1, 2, 3,  1;
  1, 3, 4,  4,  1;
  1, 3, 6,  7,  5,  1;
  1, 4, 7, 11, 11,  6, 1;
  1, 4, 9, 15, 19, 16, 7, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..200, flattened

Crossrefs

Cf. A003116 (row sums), A168396.

Programs

Maple

b:= proc(n, k) option remember; expand(`if`(n=0, 1,
      x*add(b(n-j, j), j=1..min(n, k+1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
seq(T(n), n=1..14);  # Alois P. Heinz, Jan 21 2022

Mathematica

b[n_, k_] := b[n, k] = Expand[If[n == 0, 1,
     x*Sum[b[n - j, j], {j, 1, Min[n, k + 1]}]]];
T[n_] := Rest@CoefficientList[b[n, n], x];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Apr 14 2022, after Alois P. Heinz *)