A239550 Number A(n,k) of compositions of n such that the first part is 1 and the second differences of the parts are in {-k,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 4, 4, 3, 1, 1, 1, 2, 4, 7, 6, 2, 1, 1, 1, 2, 4, 7, 11, 9, 2, 1, 1, 1, 2, 4, 8, 13, 18, 13, 3, 1, 1, 1, 2, 4, 8, 15, 23, 32, 18, 3, 1, 1, 1, 2, 4, 8, 15, 28, 40, 53, 24, 2, 1, 1, 1, 2, 4, 8, 16, 29, 52, 73, 89, 34, 3

Offset: 0

Author

Alois P. Heinz, Mar 21 2014

Keywords

• nonn
• tabl

Examples

			A(6,0) = 3: [1,1,1,1,1,1], [1,2,3], [1,5].
A(5,1) = 4: [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,4].
A(4,2) = 4: [1,1,1,1], [1,1,2], [1,2,1], [1,3].
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,  1,  1, ...
  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
  2,  3,  4,  4,  4,  4,  4,  4,  4, ...
  2,  4,  7,  7,  8,  8,  8,  8,  8, ...
  3,  6, 11, 13, 15, 15, 16, 16, 16, ...
  2,  9, 18, 23, 28, 29, 31, 31, 32, ...
  2, 13, 32, 40, 52, 56, 60, 61, 63, ...
		

Links

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

Crossrefs

Columns k=0-10 gives: A129654, A239551, A239552, A239553, A239554, A239555, A239556, A239557, A239558, A239559, A239560.

Main diagonal gives A239561.

Programs

Maple

b:= proc(n, i, j, k) option remember; `if`(n=0, 1,
      `if`(i=0, add(b(n-h, j, h, k), h=1..n), add(
       b(n-h, j, h, k), h=max(1, 2*j-i-k)..min(n, 2*j-i+k))))
    end:
A:= (n, k)-> `if`(n=0, 1, b(n-1, 0, 1, k)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

b[n_, i_, j_, k_] := b[n, i, j, k] = If[n == 0, 1, If[i == 0, Sum[b[n-h, j, h, k], {h, 1, n}], Sum[b[n-h, j, h, k], {h, Max[1, 2*j - i - k], Min[n, 2*j - i + k]}]]] ; A[n_, k_] := If[n == 0, 1, b[n-1, 0, 1, k]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 22 2015, after Alois P. Heinz *)