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A300663 Expansion of 1/(1 - Sum_{k>=1} mu(k)*x^k), where mu() is the Moebius function (A008683).

%I A300663 #16 Apr 06 2022 11:46:26
%S A300663 1,1,0,-2,-3,-2,3,8,8,-2,-16,-24,-10,24,59,54,-11,-117,-174,-90,162,
%T A300663 431,449,-20,-835,-1393,-848,1062,3352,3748,317,-6257,-11134,-7583,
%U A300663 7294,25956,30786,5217,-46545,-88132,-65062,48534,199234,249263,63034,-342174,-691679,-554002
%N A300663 Expansion of 1/(1 - Sum_{k>=1} mu(k)*x^k), where mu() is the Moebius function (A008683).
%C A300663 Invert transform of A008683.
%H A300663 Seiichi Manyama, <a href="/A300663/b300663.txt">Table of n, a(n) for n = 0..5000</a>
%H A300663 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A300663 G.f.: 1/(1 - Sum_{k>=1} A008683(k)*x^k).
%F A300663 a(0) = 1; a(n) = Sum_{k=1..n} mu(k) * a(n-k). - _Seiichi Manyama_, Apr 06 2022
%p A300663 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A300663       numtheory[mobius](j)*a(n-j), j=1..n))
%p A300663     end:
%p A300663 seq(a(n), n=0..50);  # _Alois P. Heinz_, Mar 10 2018
%t A300663 nmax = 47; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
%t A300663 a[0] = 1; a[n_] := a[n] = Sum[MoebiusMu[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 47}]
%o A300663 (PARI) my(N=66, x='x+O('x^N)); Vec(1/(1-sum(k=1, N, moebius(k)*x^k))) \\ _Seiichi Manyama_, Apr 06 2022
%o A300663 (PARI) a(n) = if(n==0, 1, sum(k=1, n, moebius(k)*a(n-k))); \\ _Seiichi Manyama_, Apr 06 2022
%Y A300663 Cf. A008683, A050385, A068341, A073776, A073777, A117209, A117210, A185694, A280194.
%K A300663 sign
%O A300663 0,4
%A A300663 _Ilya Gutkovskiy_, Mar 10 2018