A242887 Number T(n,k) of compositions of n into parts with distinct multiplicities and with exactly k parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 3, 1, 0, 1, 0, 6, 4, 1, 0, 1, 1, 4, 4, 5, 1, 0, 1, 0, 9, 8, 15, 6, 1, 0, 1, 1, 9, 5, 15, 21, 7, 1, 0, 1, 0, 10, 8, 20, 6, 28, 8, 1, 0, 1, 1, 12, 12, 6, 96, 42, 36, 9, 1, 0, 1, 0, 15, 12, 30, 192, 168, 64, 45, 10, 1, 0, 1, 1, 13, 9, 20, 142, 238, 204, 93, 55, 11, 1

Offset

0,14

Links

Alois P. Heinz, Rows n = 0..140, flattened

Example

T(5,1) = 1: [5].
T(5,3) = 6: [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], [3,1,1].
T(5,4) = 4: [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1].
T(5,5) = 1: [1,1,1,1,1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 0,  1;
  0, 1, 1,  3,  1;
  0, 1, 0,  6,  4,  1;
  0, 1, 1,  4,  4,  5,  1;
  0, 1, 0,  9,  8, 15,  6,  1;
  0, 1, 1,  9,  5, 15, 21,  7,  1;
  0, 1, 0, 10,  8, 20,  6, 28,  8, 1;
  0, 1, 1, 12, 12,  6, 96, 42, 36, 9, 1;

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, x^j*
       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_] := b[n, i, s] = If[n==0, Total[s]!, If[i<1, 0, Expand[Sum[ If[j>0 && MemberQ[s, j], 0, x^j*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 08 2017, translated from Maple *)

Prog

(PARI)
T(n)={Vecrev(((r, k, b, w)->if(!k||!r, if(r, 0, w!*x^w), sum(m=0, r\k, if(!m || !bittest(b, m), self()(r-k*m, k-1, bitor(b, 1<<m), w+m)/m!))))(n, n, 1, 0))}
{ for(n=0, 10, print(T(n))) } \\ Andrew Howroyd, Aug 31 2019

Crossrefs

Columns k=0-10 give: A000007, A057427, A059841 (for n>1), A321773, A321774, A321775, A321776, A321777, A321778, A321779, A321780.

Row sums give A242882.

T(2n,n) gives A321772.

Cf. A242896.

Sequence in context: A193002 A366725 A122960 * A307753 A181116 A051834

Adjacent sequences: A242884 A242885 A242886 * A242888 A242889 A242890

Keyword

nonn,tabl

Author

Alois P. Heinz, May 25 2014

Status

approved