A214257 Number A(n,k) of compositions of n where the difference between largest and smallest parts is <= 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, 15, 2, 1, 1, 2, 4, 8, 16, 27, 27, 4, 1, 1, 2, 4, 8, 16, 30, 47, 39, 3, 1, 1, 2, 4, 8, 16, 32, 59, 88, 63, 4, 1, 1, 2, 4, 8, 16, 32, 62, 111, 158, 100, 2

Offset: 0

Author

Alois P. Heinz, Jul 08 2012

Keywords

• nonn
• tabl

Examples

			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(5,1) =  8: [5], [3,2], [2,3], [2,2,1], [2,1,2], [2,1,1,1], [1,2,2], [1,2,1,1], [1,1,2,1], [1,1,1,2], [1,1,1,1,1],
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, 15, 27, 30, 32, 32, 32, 32, ...
  2, 27, 47, 59, 62, 64, 64, 64, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..150, flattened

Crossrefs

Columns k=0-1 give: A000005, A072951.

Main diagonal gives: A011782.

Cf. A214246, A214247, A214248, A214249, A214258, A214268, A214269.

Programs

Maple

b:= proc(n, k, s, t) option remember;
      `if`(n<0, 0, `if`(n=0, 1, add(b(n-j, k,
       min(s,j), max(t,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=1..n)):
seq(seq(A(n, d-n), n=0..d), d=0..11);
# second Maple program:
b:= proc(n, s, t) option remember; `if`(n=0, x^(t-s),
      add(b(n-j, min(s, j), max(t, j)), j=1..n))
    end:
T:= (n, k)-> coeff(b(n$2, 0), x, k):
A:= proc(n, k) option remember; `if`(k<0, 0,
      `if`(k>n, A(n$2), A(n, k-1)+T(n, k)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..11);  # Alois P. Heinz, Jan 05 2019

Mathematica

b[n_, k_, s_, t_] := b[n, k, s, t] = If[n < 0, 0, If[n == 0, 1, Sum [b[n - j, k, Min[s, j], Max[t, 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, 1, n}]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 11}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

Formula

T(n,k) = Sum_{i=0..k} A214258(n,i).