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A292561 Expansion of Product_{k>=1} (1 - mu(k)^2*x^k), where mu() is the Moebius function (A008683).

%I A292561 #10 Jul 26 2025 16:31:07
%S A292561 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 A292561 1,-10,0,13,2,-10,-3,14,-2,-17,-3,21,5,-22,0,22,4,-34,-5,33,9,-33,-10,
%U A292561 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 A292561 Expansion of Product_{k>=1} (1 - mu(k)^2*x^k), where mu() is the Moebius function (A008683).
%C A292561 Convolution inverse of A073576.
%C A292561 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 A292561 Alois P. Heinz, <a href="/A292561/b292561.txt">Table of n, a(n) for n = 0..10000</a>
%H A292561 <a href="/index/Par#part">Index entries for related partition-counting sequences</a>
%F A292561 G.f.: Product_{k>=1} (1 - x^A005117(k)).
%p A292561 with(numtheory):
%p A292561 b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A292561       abs(mobius(d)), d=divisors(j)) *b(n-j), j=1..n)/n)
%p A292561     end:
%p A292561 a:= proc(n) option remember; `if`(n=0, 1,
%p A292561       -add(b(n-i)*a(i), i=0..n-1))
%p A292561     end:
%p A292561 seq(a(n), n=0..80);  # _Alois P. Heinz_, Sep 20 2017
%t A292561 nmax = 75; CoefficientList[Series[Product[1 - MoebiusMu[k]^2 x^k, {k, 1, nmax}], {x, 0, nmax}], x]
%Y A292561 Cf. A005117, A008683, A046675, A073576, A087188.
%K A292561 sign,look
%O A292561 0,9
%A A292561 _Ilya Gutkovskiy_, Sep 19 2017