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Expansion of Product_{k>=1} (1 - mu(k)^2*x^k), where mu() is the Moebius function (A008683).
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%I #5 Sep 20 2017 20:19:27

%S 1,-1,-1,0,1,0,-1,1,2,0,-3,0,2,0,-3,0,5,0,-4,-2,4,0,-5,0,7,3,-8,-1,5,

%T 1,-10,0,13,2,-10,-3,14,-2,-17,-3,21,5,-22,0,22,4,-34,-5,33,9,-33,-10,

%U 43,6,-43,-19,52,16,-51,-13,56,24,-71,-20,64,26,-78,-24,90,24,-90,-39,112,26,-115,-37

%N Expansion of Product_{k>=1} (1 - mu(k)^2*x^k), where mu() is the Moebius function (A008683).

%C Convolution inverse of A073576.

%C The difference between the number of partitions of n into an even number of distinct squarefree parts and the number of partitions of n into an odd number of distinct squarefree parts.

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

%F G.f.: Product_{k>=1} (1 - x^A005117(k)).

%p with(numtheory):

%p b:= proc(n) option remember; `if`(n=0, 1, add(add(d*

%p abs(mobius(d)), d=divisors(j)) *b(n-j), j=1..n)/n)

%p end:

%p a:= proc(n) option remember; `if`(n=0, 1,

%p -add(b(n-i)*a(i), i=0..n-1))

%p end:

%p seq(a(n), n=0..80); # _Alois P. Heinz_, Sep 20 2017

%t nmax = 75; CoefficientList[Series[Product[1 - MoebiusMu[k]^2 x^k, {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A005117, A008683, A046675, A073576, A087188.

%K sign

%O 0,9

%A _Ilya Gutkovskiy_, Sep 19 2017