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%I #10 Aug 22 2023 11:58:44
%S 1,2,-2,-4,-2,-8,4,16,-2,14,8,-20,4,-24,-16,16,-2,36,-14,-36,8,-32,20,
%T 48,4,42,24,-40,-16,-56,-16,64,-2,40,-36,-64,-14,-72,36,48,8,84,32,
%U -84,20,-56,-48,96,4,114,-42,-72,24,-104,40,80,-16,72,56,-116,-16,-120,-64,112,-2,96,-40,-132,-36,-96,64,144,-14
%N Expansion of eta(q^2)^7*eta(q^4)/(eta(q)*eta(q^8)^2) in powers of q.
%H Amiram Eldar, <a href="/A113416/b113416.txt">Table of n, a(n) for n = 0..10000</a>
%F Euler transform of period 8 sequence [2, -5, 2, -6, 2, -5, 2, -4, ...].
%F a(n) = 2*b(n) where b(n) is multiplicative and b(2^e) = -1 if e>0, b(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 7 (mod 8), b(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 3, 5 (mod 8).
%F G.f.: 1+2(Sum_{k>0} (2k-1)*(-1)^[k/2]*x^(2k-1)/(1+x^(2k-1))) = Product_{k>0} (1-x^(2k))^4*(1+x^k)^2/((1+x^(2k))*(1+x^(4k))^2).
%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[0] = 1; a[1] = 2; a[n_] := 2 * Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* _Amiram Eldar_, Aug 22 2023 *)
%o (PARI) a(n)=if(n<=0, n==0, -2*sumdiv(n,d, d*(d%2)*(-1)^(n/d+(d+1)\4)))
%o (PARI) {a(n)=local(A,p,e); if(n<=0, n==0, A=factor(n); 2*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)))))}
%o (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^7*eta(x^4+A)/(eta(x+A)*eta(x^8+A))^2, n))}
%K sign
%O 0,2
%A _Michael Somos_, Oct 29 2005
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