A242882 Number of compositions of n into parts with distinct multiplicities.

1, 1, 2, 2, 6, 12, 16, 40, 60, 82, 216, 538, 788, 2034, 3740, 6320, 13336, 27498, 42936, 93534, 173520, 351374, 734650, 1592952, 3033194, 6310640, 12506972, 25296110, 49709476, 101546612, 195037028, 391548336, 764947954, 1527004522, 2953533640, 5946359758

Offset: 0

Author

Alois P. Heinz, May 25 2014

Keywords

• nonn

Examples

			a(0) = 1: the empty composition.
a(1) = 1: [1].
a(2) = 2: [1,1], [2].
a(3) = 2: [1,1,1], [3].
a(4) = 6: [1,1,1,1], [1,1,2], [1,2,1], [2,1,1], [2,2], [4].
a(5) = 12: [1,1,1,1,1], [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], [5].
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..264 (terms 0..200 from Alois P. Heinz)
• Vaclav Kotesovec, What is the limit a(n)/2^n ?

Crossrefs

Row sums of A242887 and of A242896.

Cf. A098859 (the same for partitions).

Programs

Maple

b:= proc(n, i, s) option remember; `if`(n=0, add(j, j=s)!,
      `if`(i<1, 0, add(`if`(j>0 and j in s, 0,
      b(n-i*j, i-1, `if`(j=0, s, s union {j}))/j!), j=0..n/i)))
    end:
a:= n-> b(n$2, {}):
seq(a(n), n=0..45);

Mathematica

b[n_, i_, s_] := b[n, i, s] = If[n == 0, Sum[j, {j, s}]!, If[i < 1, 0, Sum[If[j > 0 && MemberQ[s, j], 0, b[n - i*j, i - 1, If[j == 0, s, s ~Union~ {j}]]/j!], {j, 0, n/i}]]];
a[n_] := b[n, n, {}];
Table[a[n], {n, 0, 45}] (* Jean-François Alcover, May 17 2018, translated from Maple *)

PARI

a(n)={((r,k,b,w)->if(!k||!r, if(r,0,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