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A351407 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(n/d) * a(d).

%I A351407 #5 Feb 10 2022 10:03:21
%S A351407 1,-1,2,-3,3,-4,8,-9,6,-9,14,-15,16,-17,27,-33,21,-22,36,-37,34,-45,
%T A351407 61,-62,51,-55,73,-82,76,-77,124,-125,80,-97,120,-132,132,-133,171,
%U A351407 -190,153,-154,221,-222,194,-233,296,-297,239,-248,313,-337,301,-302
%N A351407 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(n/d) * a(d).
%F A351407 G.f. A(x) satisfies: A(x) = x * ( 1 - A(x) + A(x^2) - A(x^3) + A(x^4) - A(x^5) + ... ).
%F A351407 G.f.: x * ( 1 - Sum_{n>=1} a(n) * x^n / (1 + x^n) ).
%p A351407 a:= proc(n) option remember; `if`(n=1, 1,
%p A351407       add((-1)^((n-1)/d)*a(d), d=numtheory[divisors](n-1)))
%p A351407     end:
%p A351407 seq(a(n), n=1..54);  # _Alois P. Heinz_, Feb 10 2022
%t A351407 a[1] = 1; a[n_] := a[n] = Sum[(-1)^((n - 1)/d) a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 54}]
%t A351407 nmax = 54; A[_] = 0; Do[A[x_] = x (1 + Sum[(-1)^k A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) //Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
%Y A351407 Cf. A003238, A067856, A281487, A307776, A307778, A308077, A325144, A343370.
%K A351407 sign
%O A351407 1,3
%A A351407 _Ilya Gutkovskiy_, Feb 10 2022