A307815 Number of partitions of n into 3 squarefree parts.

0, 0, 0, 1, 1, 2, 2, 3, 3, 5, 4, 5, 5, 7, 7, 9, 8, 11, 11, 13, 11, 15, 14, 18, 15, 20, 19, 23, 20, 24, 24, 27, 24, 30, 29, 34, 30, 37, 36, 42, 36, 45, 44, 50, 44, 54, 54, 59, 52, 62, 63, 68, 57, 69, 70, 78, 65, 78, 78, 88, 74, 86, 87, 98, 84, 98, 98, 107, 93, 109, 108, 120, 102, 124, 123

Offset

0,6

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Rémy Sigrist)

Index entries for sequences related to partitions

Formula

a(n) = [x^n y^3] Product_{k>=1} 1/(1 - mu(k)^2*y*x^k).

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} mu(i)^2 * mu(k)^2 * mu(n-i-k)^2, where mu is the Mobius function. - Wesley Ivan Hurt, May 09 2019

Example

a(10) = 4 because we have [7, 2, 1], [6, 3, 1], [6, 2, 2] and [5, 3, 2].

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0$3], `if`(i<1, 0, b(n, i-1)+
      `if`(numtheory[issqrfree](i), [0, b(n-i, min(i, n-i))[1..3][]], 0)))
    end:
a:= n-> b(n$2)[4]:
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 30 2019

Mathematica

Array[Count[IntegerPartitions[#, {3}], _?(AllTrue[#, SquareFreeQ] &)] &, 75, 0]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, {1, 0, 0, 0}, If[i < 1, {0, 0, 0, 0}, b[n, i - 1] + If[SquareFreeQ[i], {0, Sequence @@ b[n - i, Min[i, n - i]][[1 ;; 3]]}, {0, 0, 0, 0}]]];
a[n_] := b[n, n][[4]];
a /@ Range[0, 80] (* Jean-François Alcover, Jun 06 2021, after Alois P. Heinz *)

Crossrefs

Cf. A005117, A008683, A068307, A071068, A073576, A098235, A280210, A307727.

Sequence in context: A113605 A369984 A342515 * A070230 A007150 A213634

Adjacent sequences: A307812 A307813 A307814 * A307816 A307817 A307818

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 30 2019

Status

approved