A214249 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,...,k} \ {0}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 4, 4, 1, 1, 1, 1, 3, 4, 5, 5, 1, 1, 1, 1, 3, 4, 7, 11, 5, 1, 1, 1, 1, 3, 4, 7, 12, 14, 7, 1, 1, 1, 1, 3, 4, 7, 14, 20, 18, 10, 1, 1, 1, 1, 3, 4, 7, 14, 21, 30, 36, 9, 1, 1, 1, 1, 3, 4, 7, 14, 23, 36, 50, 49, 14, 1

Offset: 0

Author

Alois P. Heinz, Jul 08 2012

Keywords

• nonn
• tabl

Examples

			A(3,0) = 1: [3].
A(4,1) = 2: [4], [1,2,1].
A(5,2) = 5: [5], [3,2], [2,3], [2,1,2], [1,3,1].
A(6,3) = 12: [6], [4,2], [3,2,1], [3,1,2], [2,4], [2,3,1], [2,1,3], [2,1,2,1], [1,4,1], [1,3,2], [1,2,3], [1,2,1,2].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  3,  3,  3,  3,  3,  3,  3, ...
  1,  2,  4,  4,  4,  4,  4,  4, ...
  1,  4,  5,  7,  7,  7,  7,  7, ...
  1,  5, 11, 12, 14, 14, 14, 14, ...
  1,  5, 14, 20, 21, 23, 23, 23, ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Columns k=0-2 give: A000012, A173258, A214256.

Main diagonal gives: A003242.

Cf. A214246, A214247, A214248, A214257, A214258, A214268, A214269.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n<1 or i<1, 0,
      `if`(n=i, 1, add(b(n-i, i+j, k), j={$-k..k} minus{0})))
    end:
A:= (n, k)-> `if`(n=0, 1, add(b(n, j, min(n, k)), j=1..n)):
seq(seq(A(n, d-n), n=0..d), d=0..15);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n<1 || i<1, 0, If[n == i, 1, Sum[b[n-i, i+j, k], {j, Range[-k, -1] ~Join~ Range[k]}]]]; A[n_, k_] := If[n == 0, 1, Sum[b[n, j, Min[n, k]], {j, 1, n}]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)