%I #14 Jan 28 2024 02:00:33
%S 1,-1,-2,-1,-4,2,8,-1,7,4,-10,2,-12,-8,8,-1,18,-7,-18,4,-16,10,24,2,
%T 21,12,-20,-8,-28,-8,32,-1,20,-18,-32,-7,-36,18,24,4,42,16,-42,10,-28,
%U -24,48,2,57,-21,-36,12,-52,20,40,-8,36,28,-58,-8,-60,-32,56,-1,48,-20,-66,-18,-48,32,72,-7,74,36,-42,18,-80,-24
%N Expansion of (eta(q^2)^7*eta(q^4)/(eta(q)*eta(q^8))^2-1)/2 in powers of q.
%H Amiram Eldar, <a href="/A113418/b113418.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) is multiplicative and a(2^e) = -1 if e>0, a(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 7 (mod 8), a(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 3, 5 (mod 8).
%F G.f.: Sum_{k>0} (2k-1)*(-1)^[k/2]*x^(2k-1)/(1+x^(2k-1)).
%F From _Amiram Eldar_, Jan 28 2024: (Start)
%F a(n) = (-1)^(n+1) * A117000(n).
%F Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(2)) = 0.290786... . (End)
%t f[p_, e_] := If[1 < Mod[p, 8] < 7, ((-p)^(e+1)-1)/(-p-1), (p^(e+1)-1)/(p-1)]; f[2, e_] := -1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Aug 22 2023 *)
%o (PARI) a(n)=if(n<1, 0, -sumdiv(n,d, d*(d%2)*(-1)^(n/d+(d+1)\4)))
%o (PARI) {a(n)=local(A,p,e); if(n<1, 0, A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==2, -1, p*=kronecker(2,p); (p^(e+1)-1)/(p-1)))))}
%Y Apart from signs, same as A117000.
%Y A113416(n)=2*a(n) if n>0.
%K sign,mult
%O 1,3
%A _Michael Somos_, Oct 29 2005