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Expansion of Product_{k>=1} (1 + mu(k)*x^k).
2

%I #6 Oct 02 2022 10:28:59

%S 1,1,-1,-2,-1,0,1,1,0,0,1,0,-2,-2,1,4,3,-2,-4,-2,0,2,3,0,-1,1,0,-3,-3,

%T -1,2,4,3,0,-2,-1,2,0,-5,-3,3,3,0,-2,-4,-2,4,5,3,3,1,-4,-9,-8,3,11,6,

%U 0,-3,-7,-4,2,-1,-2,6,8,-2,-10,-8,4,14,11,2,-6,-11,-5

%N Expansion of Product_{k>=1} (1 + mu(k)*x^k).

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

%t a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Sum[d (-MoebiusMu[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

%Y Cf. A008683, A087188, A117210, A185694, A300663, A306327, A357521, A357524.

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Oct 02 2022