A286141 Number of partitions of n into a squarefree number of parts.

1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 30, 40, 53, 70, 92, 120, 154, 199, 254, 324, 409, 517, 648, 811, 1008, 1253, 1549, 1911, 2347, 2880, 3519, 4294, 5219, 6338, 7671, 9273, 11173, 13451, 16147, 19359, 23151, 27656, 32958, 39231, 46594, 55276, 65444, 77391, 91341, 107689, 126734

Offset: 0

Author

Ilya Gutkovskiy, May 07 2017

Keywords

• nonn

Comments

Also number of partitions of n such that the largest part is a squarefree (A005117).

Examples

			a(6) = 9 because we have [6], [5, 1], [4, 2], [4, 1, 1], [3, 3], [3, 2, 1], [2, 2, 2], [2, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1] (partitions into a squarefree number of parts).
Also a(6) = 9 because we have [6], [5, 1], [3, 3], [3, 2, 1], [3, 1, 1, 1], [2, 2, 2], [2, 2, 1, 1], [2, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1] (partitions such that the largest part is a squarefree).
		

Links

• Index entries for related partition-counting sequences

Crossrefs

Cf. A005117, A038499, A073576, A089333, A102848, A178927.

Programs

Mathematica

Join[{1}, Table[Length@Select[IntegerPartitions@n, SquareFreeQ@Length@# &], {n, 50}]]
nmax = 50; CoefficientList[Series[1 + Sum[MoebiusMu[i]^2 x^i/Product[1 - x^j, {j, 1, i}], {i, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: 1 + Sum_{i>=1} x^A005117(i) / Product_{j=1..A005117(i)} (1 - x^j).