A327549 Number T(n,k) of compositions of partitions of n with exactly k compositions; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 1, 0, 4, 2, 1, 0, 8, 8, 2, 1, 0, 16, 16, 8, 2, 1, 0, 32, 48, 24, 8, 2, 1, 0, 64, 96, 64, 24, 8, 2, 1, 0, 128, 256, 160, 80, 24, 8, 2, 1, 0, 256, 512, 448, 192, 80, 24, 8, 2, 1, 0, 512, 1280, 1024, 576, 224, 80, 24, 8, 2, 1

Offset: 0

Author

Alois P. Heinz, Sep 16 2019

Keywords

• nonn
• tabl

Examples

			T(3,1) = 4: 3, 21, 12, 111.
T(3,2) = 2: 2|1, 11|1.
T(3,3) = 1: 1|1|1.
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,    1;
  0,   4,    2,    1;
  0,   8,    8,    2,   1;
  0,  16,   16,    8,   2,   1;
  0,  32,   48,   24,   8,   2,  1;
  0,  64,   96,   64,  24,   8,  2,  1;
  0, 128,  256,  160,  80,  24,  8,  2, 1;
  0, 256,  512,  448, 192,  80, 24,  8, 2, 1;
  0, 512, 1280, 1024, 576, 224, 80, 24, 8, 2, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Columns k=0-2 give: A000007, A011782 (for n>0), A134353(n-2) (for n>1).

Row sums give A075900.

T(2n,n) gives A327550.

Cf. A060642, A327548.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+expand(2^(i-1)*x*b(n-i, min(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..12);

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + 2^(i-1) x b[n-i, Min[n-i, i]]]];
T[n_] := CoefficientList[b[n, n], x];
T /@ Range[0, 12] // Flatten (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

Formula

Sum_{k=1..n} k * T(n,k) = A327548(n).