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