A242896 Number T(n,k) of compositions of n into k parts with distinct multiplicities, where parts are counted without multiplicities; triangle T(n,k), n>=0, 0<=k<=max{i:A000292(i)<=n}, read by rows.

1, 0, 1, 0, 2, 0, 2, 0, 3, 3, 0, 2, 10, 0, 4, 12, 0, 2, 38, 0, 4, 56, 0, 3, 79, 0, 4, 152, 60, 0, 2, 251, 285, 0, 6, 284, 498, 0, 2, 594, 1438, 0, 4, 920, 2816, 0, 4, 1108, 5208, 0, 5, 2136, 11195, 0, 2, 3402, 24094, 0, 6, 4407, 38523, 0, 2, 8350, 85182

Offset: 0

Author

Alois P. Heinz, May 25 2014

Keywords

• nonn
• tabf

Examples

			T(5,1) = 2: [1,1,1,1,1], [5].
T(5,2) = 10: [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1], [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], [3,1,1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 2;
  0, 2;
  0, 3,   3;
  0, 2,  10;
  0, 4,  12;
  0, 2,  38;
  0, 4,  56;
  0, 3,  79;
  0, 4, 152, 60;
		

Links

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

Crossrefs

Columns k=0-7 give: A000007, A000005, A242900, A246230, A246231, A246232, A246233, A246234.

Row sums give A242882.

Cf. A182485 (the same for partitions), A242887.

Programs

Maple

b:= proc(n, i, s) option remember; `if`(n=0, add(j, j=s)!,
      `if`(i<1, 0, expand(add(`if`(j>0 and j in s, 0, `if`(j=0, 1, x)*
       b(n-i*j, i-1, `if`(j=0, s, s union {j}))/j!), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, {})):
seq(T(n), n=0..16);

Mathematica

b[n_, i_, s_List] := b[n, i, s] = If[n == 0, Total[s]!, If[i<1, 0, Expand[ Sum[ If[j>0 && MemberQ[s, j], 0, If[j == 0, 1, x]*b[n-i*j, i-1, If[j == 0, s, s ~Union~ {j}]]/j!], {j, 0, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, {}]]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)