A214258 Number T(n,k) of compositions of n where the difference between largest and smallest parts equals k; triangle T(n,k), n>=1, 0<=k<n, read by rows.

1, 2, 0, 2, 2, 0, 3, 3, 2, 0, 2, 9, 3, 2, 0, 4, 11, 12, 3, 2, 0, 2, 25, 20, 12, 3, 2, 0, 4, 35, 49, 23, 12, 3, 2, 0, 3, 60, 95, 58, 23, 12, 3, 2, 0, 4, 96, 188, 123, 61, 23, 12, 3, 2, 0, 2, 157, 366, 266, 132, 61, 23, 12, 3, 2, 0, 6, 241, 714, 557, 294, 135, 61, 23, 12, 3, 2, 0

Offset

1,2

Comments

For fixed k > 0, limit_{n->infinity} T(n,k)^(1/n) = d, where d > 1 is the real root of the equation d^(k+2) - 2*d^(k+1) + 1 = 0. - Vaclav Kotesovec, Jan 07 2019

Links

Alois P. Heinz, Rows n = 1..150, flattened

Formula

T(n,0) = A214257(n,0), T(n,k) = A214257(n,k)-A214257(n,k-1) for k>0.

Example

T(4,0) = 3: [4], [2,2], [1,1,1,1].
T(5,1) = 9: [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].
T(5,2) = 3: [3,1,1], [1,3,1], [1,1,3].
T(5,3) = 2: [4,1], [1,4].
T(6,2) = 12: [4,2], [3,2,1], [3,1,2], [3,1,1,1], [2,4], [2,3,1], [2,1,3], [1,3,2], [1,3,1,1], [1,2,3], [1,1,3,1], [1,1,1,3].
Triangle T(n,k) begins:
  1;
  2,  0;
  2,  2,  0;
  3,  3,  2,  0;
  2,  9,  3,  2,  0;
  4, 11, 12,  3,  2,  0;
  2, 25, 20, 12,  3,  2,  0;
  4, 35, 49, 23, 12,  3,  2,  0;

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:= proc(n, k) option remember;
      `if`(n=0, 1, add(b(n-j, k+1, j, j), j=1..n))
    end:
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n-1), n=1..15);
# 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-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n$2, 0)):
seq(T(n), n=1..15);  # 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_] := A[n, k] = If[n == 0, 1, Sum[b[n-j, k+1, j, j], {j, 1, n}]]; T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)

Crossrefs

Columns k=0-10 give: A000005, A214259, A323119, A323120, A323121, A323122, A323123, A323124, A323125, A323126, A323127.

Row sums give: A011782.

T(2n,n) gives A323111.

Cf. A214246, A214247, A214248, A214249, A214257, A214268, A214269.

Sequence in context: A208955 A121363 A346274 * A369279 A138021 A166065

Adjacent sequences: A214255 A214256 A214257 * A214259 A214260 A214261

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 08 2012

Status

approved