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

Author

Alois P. Heinz, May 14 2014

Keywords

• nonn
• look

Comments

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

Examples

			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].
		

Links

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

Crossrefs

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

Programs

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 *)