A322428 Sum T(n,k) of k-th largest parts of all compositions of n; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 3, 1, 8, 3, 1, 19, 8, 4, 1, 43, 20, 11, 5, 1, 94, 48, 27, 16, 6, 1, 202, 110, 64, 42, 22, 7, 1, 428, 245, 149, 100, 64, 29, 8, 1, 899, 533, 341, 228, 163, 93, 37, 9, 1, 1875, 1142, 765, 512, 383, 256, 130, 46, 10, 1, 3890, 2420, 1683, 1144, 859, 638, 386, 176, 56, 11, 1

Offset

1,2

Links

Alois P. Heinz, Rows n = 1..50, flattened

Example

The 4 compositions of 3 are: 111, 12, 21, 3.  The sums of k-th largest parts for k=1..3 give: 1+2+2+3 = 8, 1+1+1+0 = 3, 1+0+0+0 = 1.
Triangle T(n,k) begins:
     1;
     3,    1;
     8,    3,   1;
    19,    8,   4,   1;
    43,   20,  11,   5,   1;
    94,   48,  27,  16,   6,   1;
   202,  110,  64,  42,  22,   7,   1;
   428,  245, 149, 100,  64,  29,   8,  1;
   899,  533, 341, 228, 163,  93,  37,  9,  1;
  1875, 1142, 765, 512, 383, 256, 130, 46, 10, 1;
  ...

Maple

b:= proc(n, l) option remember; `if`(n=0, add(l[-i]*x^i,
      i=1..nops(l)), add(b(n-j, sort([l[], j])), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n, [])):
seq(T(n), n=1..12);

Mathematica

b[n_, l_] := b[n, l] = If[n == 0, Sum[l[[-i]] x^i, {i, 1, Length[l]}], Sum[b[n - j, Sort[Append[l, j]]], {j, 1, n}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][ b[n, {}]];
Array[T, 12] // Flatten (* Jean-François Alcover, Dec 29 2018, after Alois P. Heinz *)

Crossrefs

Column k=1 gives A102712.

Row sums give A001787.

T(n+1,1+ceiling(n/2)) gives A027306.

Cf. A322427.

Sequence in context: A352939 A249757 A207609 * A130300 A366873 A345656

Adjacent sequences: A322425 A322426 A322427 * A322429 A322430 A322431

Keyword

nonn,tabl

Author

Alois P. Heinz, Dec 07 2018

Status

approved