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%I #13 Mar 30 2023 15:12:15
%S 1,-1,1,0,1,-3,1,0,2,-3,1,2,1,-3,3,0,1,-8,1,2,3,-3,1,0,2,-3,4,2,1,-13,
%T 1,0,3,-3,3,10,1,-3,3,0,1,-13,1,2,8,-3,1,0,2,-8,3,2,1,-20,3,0,3,-3,1,
%U 18,1,-3,8,0,3,-13,1,2,3,-13,1,-4,1,-3,8,2,3,-13,1
%N G.f. A(x) satisfies: A(x) = x - A(x^2) + A(x^3) - A(x^4) + A(x^5) - A(x^6) + ...
%H Seiichi Manyama, <a href="/A308077/b308077.txt">Table of n, a(n) for n = 1..10000</a>
%F a(1) = 1; a(n) = Sum_{d|n, d<n} (-1)^(n/d+1)*a(d).
%p a:= proc(n) option remember; `if`(n<2, n, add(a(n/d)*
%p (-1)^(d-1), d=numtheory[divisors](n) minus {1}))
%p end:
%p seq(a(n), n=1..80); # _Alois P. Heinz_, Mar 30 2023
%t terms = 79; A[_] = 0; Do[A[x_] = x + Sum[(-1)^(k + 1) A[x^k], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
%t a[n_] := If[n == 1, n, Sum[If[d < n, (-1)^(n/d + 1) a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 79}]
%Y Cf. A006005 (positions of 1's), A067856, A307776, A347031.
%K sign
%O 1,6
%A _Ilya Gutkovskiy_, May 11 2019
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