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

Author

Alois P. Heinz, May 25 2014

Keywords

• nonn
• tabl

Examples

			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;
		

Links

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

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.

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, 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 *)

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<Andrew Howroyd, Aug 31 2019