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

0,3

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 ?

Example

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].

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 *)

Prog

(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<<m), w+m)/m!))))(n, n, 1, 0)} \\ Andrew Howroyd, Aug 31 2019

Crossrefs

Row sums of A242887 and of A242896.

Cf. A098859 (the same for partitions).

Sequence in context: A262501 A225422 A290518 * A361426 A157285 A320068

Adjacent sequences: A242879 A242880 A242881 * A242883 A242884 A242885

Keyword

nonn

Author

Alois P. Heinz, May 25 2014

Status

approved