A214248 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 4, 3, 1, 1, 2, 4, 6, 2, 1, 1, 2, 4, 8, 11, 4, 1, 1, 2, 4, 8, 14, 17, 2, 1, 1, 2, 4, 8, 16, 27, 29, 4, 1, 1, 2, 4, 8, 16, 30, 49, 47, 3, 1, 1, 2, 4, 8, 16, 32, 59, 92, 78, 4, 1, 1, 2, 4, 8, 16, 32, 62, 113, 170, 130, 2

Offset

0,6

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Example

A(3,0) = 2: [3], [1,1,1].
A(4,1) = 6: [4], [2,2], [2,1,1], [1,2,1], [1,1,2], [1,1,1,1].
A(5,2) = 14: [5], [3,2], [3,1,1], [2,3], [2,2,1], [2,1,2], [2,1,1,1], [1,3,1], [1,2,2], [1,2,1,1], [1,1,3], [1,1,2,1], [1,1,1,2], [1,1,1,1,1].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, ...
  2,  2,  2,  2,  2,  2,  2,  2, ...
  2,  4,  4,  4,  4,  4,  4,  4, ...
  3,  6,  8,  8,  8,  8,  8,  8, ...
  2, 11, 14, 16, 16, 16, 16, 16, ...
  4, 17, 27, 30, 32, 32, 32, 32, ...
  2, 29, 49, 59, 62, 64, 64, 64, ...

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})))
    end:
A:= (n, k)-> `if`(n=0, 1, add(b(n, j, 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, -k, k}]]]; A[n_, k_] := If[n == 0, 1, Sum[b[n, j, k], {j, 1, n}]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

Crossrefs

Columns k=0-2 give: A000005, A034297, A214255.

Main diagonal gives: A011782.

Cf. A214246, A214247, A214249, A214257, A214258, A214268, A214269.

Sequence in context: A099172 A214246 A214257 * A152719 A107044 A141591

Adjacent sequences: A214245 A214246 A214247 * A214249 A214250 A214251

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 08 2012

Status

approved