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

0,5

Links

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

Example

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;

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 *)

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.

Sequence in context: A319071 A316432 A046522 * A240183 A112631 A158706

Adjacent sequences: A242893 A242894 A242895 * A242897 A242898 A242899

Keyword

nonn,tabf

Author

Alois P. Heinz, May 25 2014

Status

approved