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

0,14

Links

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

Example

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, ...

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 *)

Crossrefs

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

Main diagonal gives: A003242.

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

Sequence in context: A330827 A010276 A214268 * A271714 A049639 A046555

Adjacent sequences: A214246 A214247 A214248 * A214250 A214251 A214252

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 08 2012

Status

approved