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

Author

Alois P. Heinz, Dec 07 2018

Keywords

• nonn
• tabl

Examples

			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;
  ...
		

Links

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

Crossrefs

Column k=1 gives A097939.

Row sums give A001787.

Cf. A322428.

Programs

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 *)