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

0,5

Links

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

Formula

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

Example

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;
  ...

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 *)

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.

Sequence in context: A355756 A140649 A290222 * A293808 A327805 A276689

Adjacent sequences: A327546 A327547 A327548 * A327550 A327551 A327552

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 16 2019

Status

approved