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A307781 a(1) = 1; a(n+1) = -Sum_{d|n} a(d)^(n/d).

%I A307781 #12 May 01 2019 05:42:45
%S A307781 1,-1,0,-1,-1,0,0,-1,-2,1,-2,1,-2,1,-1,1,-5,4,-8,7,-10,9,-13,12,-15,
%T A307781 15,-19,26,-28,27,-30,29,-34,41,-66,66,-100,99,-163,170,-223,222,-323,
%U A307781 322,-420,453,-622,621,-771,770,-997,1121,-1363,1362,-1851,1883,-2562,3073,-3857,3856
%N A307781 a(1) = 1; a(n+1) = -Sum_{d|n} a(d)^(n/d).
%H A307781 Robert Israel, <a href="/A307781/b307781.txt">Table of n, a(n) for n = 1..10000</a>
%F A307781 L.g.f.: log(Product_{n>=1} (1 - a(n)*x^n)^(1/n)) = Sum_{n>=1} a(n+1)*x^n/n.
%p A307781 f:= proc(n) option remember; local d;
%p A307781   -add(procname(d)^((n-1)/d), d = numtheory:-divisors(n-1))
%p A307781 end proc:
%p A307781 f(1):= 1:
%p A307781 map(f, [$1..100]); # _Robert Israel_, Apr 29 2019
%t A307781 a[n_] := a[n] = -Sum[a[d]^((n - 1)/d), {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 60}]
%o A307781 (PARI) lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = -sumdiv(n-1, d, va[d]^((n-1)/d));); va;} \\ _Michel Marcus_, Apr 30 2019
%Y A307781 Cf. A127525, A307780.
%K A307781 sign
%O A307781 1,9
%A A307781 _Ilya Gutkovskiy_, Apr 28 2019