A286141 - OEIS

%I #22 May 07 2017 21:37:42

%S 1,1,2,3,4,6,9,12,16,22,30,40,53,70,92,120,154,199,254,324,409,517,

%T 648,811,1008,1253,1549,1911,2347,2880,3519,4294,5219,6338,7671,9273,

%U 11173,13451,16147,19359,23151,27656,32958,39231,46594,55276,65444,77391,91341,107689,126734

%N Number of partitions of n into a squarefree number of parts.

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

%H <a href="/index/Par#part">Index entries for related partition-counting sequences</a>

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

%e 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).

%e 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).

%t Join[{1}, Table[Length@Select[IntegerPartitions@n, SquareFreeQ@Length@# &], {n, 50}]]

%t 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]

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

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 07 2017