A322427 Sum T(n,k) of k-th smallest parts of all compositions of n; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 3, 1, 6, 5, 1, 12, 12, 7, 1, 22, 28, 20, 9, 1, 42, 54, 54, 30, 11, 1, 79, 106, 115, 92, 42, 13, 1, 151, 200, 239, 218, 144, 56, 15, 1, 291, 376, 471, 486, 378, 212, 72, 17, 1, 566, 708, 904, 1014, 908, 612, 298, 90, 19, 1, 1106, 1346, 1709, 2030, 2014, 1584, 939, 404, 110, 21, 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 smallest parts for k=1..3 give: 1+1+1+3 = 6, 1+2+2+0 = 5, 1+0+0+0 = 1.
Triangle T(n,k) begins:
    1;
    3,   1;
    6,   5,   1;
   12,  12,   7,    1;
   22,  28,  20,    9,   1;
   42,  54,  54,   30,  11,   1;
   79, 106, 115,   92,  42,  13,   1;
  151, 200, 239,  218, 144,  56,  15,  1;
  291, 376, 471,  486, 378, 212,  72, 17,  1;
  566, 708, 904, 1014, 908, 612, 298, 90, 19, 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 A097939.

Row sums give A001787.

Cf. A322428.

Sequence in context: A153641 A133545 A210214 * A209149 A343062 A210602

Adjacent sequences: A322424 A322425 A322426 * A322428 A322429 A322430

Keyword

nonn,tabl

Author

Alois P. Heinz, Dec 07 2018

Status

approved