A242434 Number of compositions of n in which each part p has multiplicity p.

1, 1, 0, 0, 1, 3, 0, 0, 0, 1, 4, 0, 0, 10, 60, 0, 1, 5, 0, 0, 15, 105, 0, 0, 0, 36, 286, 0, 0, 1281, 12768, 0, 0, 0, 56, 504, 1, 7, 2520, 27720, 28, 378, 1260, 0, 0, 7014, 84000, 0, 0, 4621, 83168, 360360, 210, 2346, 2522880, 37837800, 13860, 180180, 120, 1320

Offset

0,6

Comments

a(n) = 0 for n in {A001422}, a(n) > 0 for n in {A003995}.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..4500

Example

a(0) = 1: the empty composition.
a(1) = 1: [1].
a(4) = 1: [2,2].
a(5) = 3: [1,2,2], [2,1,2], [2,2,1].
a(9) = 1: [3,3,3].
a(10) = 4: [1,3,3,3], [3,1,3,3], [3,3,1,3], [3,3,3,1].
a(13) = 10: [2,2,3,3,3], [2,3,2,3,3], [2,3,3,2,3], [2,3,3,3,2], [3,2,2,3,3], [3,2,3,2,3], [3,2,3,3,2], [3,3,2,2,3], [3,3,2,3,2], [3,3,3,2,2].

Maple

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
       b(n, i-1, p) +`if`(i^2>n, 0, b(n-i^2, i-1, p+i)/i!)))
    end:
a:= n-> b(n, isqrt(n), 0):
seq(a(n), n=0..100);

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, b[n, i-1, p] + If[i^2 >n, 0, b[n-i^2, i-1, p+i]/i!]]]; a[n_] := b[n, Floor[Sqrt[n]], 0]; Table[ a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 08 2017, translated from Maple *)

Crossrefs

Cf. A033461 (the same for partitions), A336269.

Sequence in context: A328969 A027185 A035641 * A036873 A081130 A358623

Adjacent sequences: A242431 A242432 A242433 * A242435 A242436 A242437

Keyword

nonn,look

Author

Alois P. Heinz, May 14 2014

Status

approved