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 A113418 Expansion of (eta(q^2)^7*eta(q^4)/(eta(q)*eta(q^8))^2-1)/2 in powers of q. 1

%I

%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.

%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)).

%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

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Last modified March 24 00:12 EDT 2019. Contains 321444 sequences. (Running on oeis4.)