A214268 Number A(n,k) of compositions of n where the difference between largest and smallest parts is <= k and adjacent parts are unequal; 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, 3, 1, 1, 1, 1, 3, 4, 7, 11, 5, 1, 1, 1, 1, 3, 4, 7, 12, 12, 3, 1, 1, 1, 1, 3, 4, 7, 14, 20, 16, 5, 1, 1, 1, 1, 3, 4, 7, 14, 21, 28, 30, 5, 1

Offset: 0

Author

Alois P. Heinz, Jul 09 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].
A(7,4) = 21: [7], [5,2], [4,3], [4,2,1], [4,1,2], [3,4], [3,1,3], [3,1,2,1], [2,5], [2,4,1], [2,3,2], [2,1,4], [2,1,3,1], [1,5,1], [1,4,2], [1,3,2,1], [1,3,1,2], [1,2,4], [1,2,3,1], [1,2,1,3], [1,2,1,2,1].
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,  3, 11, 12, 14, 14, 14, 14, ...
  1,  5, 12, 20, 21, 23, 23, 23, ...
		

Links

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

Crossrefs

Columns k=0, 1 give: A000012, 1+A214270(n).

Main diagonal gives: A003242.

Cf. A214246, A214247, A214248, A214249, A214257, A214258, A214269.

Programs

Maple

b:= proc(n, k, s, t, l) option remember;
      `if`(n<0, 0, `if`(n=0, 1, add(`if`(j=l, 0, b(n-j, k,
       min(s, j), max(t, j), j)), j=max(1, t-k+1)..s+k-1)))
    end:
A:= (n, k)-> `if`(n=0, 1, add(b(n-j, k+1, j, j, j), j=1..n)):
seq(seq(A(n,d-n), n=0..d), d=0..14);

Mathematica

b[n_, k_, s_, t_, l_] := b[n, k, s, t, l] = If[n < 0, 0, If[n == 0, 1, Sum[If[j == l, 0, b[n - j, k, Min[s, j], Max[t, j], j]], {j, Max[1, t - k + 1], s + k - 1}]]]; A[n_, k_] := If[n == 0, 1, Sum[b[n - j, k + 1, j, j, j], {j, 1, n}]]; Table[Table[A [n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)