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A286141 Number of partitions of n into a squarefree number of parts. 1
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 (list; graph; refs; listen; history; text; internal format)
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 A018550

Adjacent sequences:  A286138 A286139 A286140 * A286142 A286143 A286144

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, May 07 2017

STATUS

approved

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Last modified January 24 13:24 EST 2020. Contains 331193 sequences. (Running on oeis4.)