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

0,3

Comments

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

Links

Table of n, a(n) for n=0..50.

Index entries for related partition-counting sequences

Formula

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

Example

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).

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]

Crossrefs

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

Sequence in context: A280424 A083365 A001935 * A007604 A013950 A350842

Adjacent sequences: A286138 A286139 A286140 * A286142 A286143 A286144

Keyword

nonn

Author

Ilya Gutkovskiy, May 07 2017

Status

approved