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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. 8
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

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

FORMULA

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

EXAMPLE

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, ...

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-Fran├žois Alcover, Dec 27 2013, translated from Maple *)

CROSSREFS

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

Main diagonal gives: A011782.

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

Sequence in context: A103444 A099172 A214246 * A214248 A152719 A107044

Adjacent sequences:  A214254 A214255 A214256 * A214258 A214259 A214260

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 08 2012

STATUS

approved

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Last modified July 6 08:51 EDT 2020. Contains 335476 sequences. (Running on oeis4.)