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%I #15 Jan 29 2024 01:19:44
%S 1,1,1,1,-4,1,-6,1,1,-4,12,1,14,-6,-4,1,-16,1,-18,-4,-6,12,24,1,21,14,
%T 1,-6,-28,-4,-30,1,12,-16,24,1,38,-18,14,-4,-40,-6,-42,12,-4,24,48,1,
%U 43,21,-16,14,-52,1,-48,-6,-18,-28,60,-4,62,-30,-6,1,-56,12,-66,-16,24,24,72,1,74,38,21,-18,-72,14,-78,-4,1
%N Expansion of 1-eta(q)*eta(q^3)*(eta(q^4)*eta(q^6))^2/eta(q^12)^2 in powers of q.
%D Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.68).
%F Multiplicative with a(2^e)=a(3^e)=1, a(p^e)=(p^(e+1)-1)/(p-1) if p = 1, 11 (mod 12), a(p^e)=((-p)^(e+1)-1)/(-p-1) if p = 5, 7 (mod 12).
%F G.f.: 1-Product_{k>0} (1-x^k)(1-x^(3k))(1-x^(4k))^2/(1+x^(6k))^2 = Sum_{k>0} x^k*(1-3*x^(2k)+x^(4k))*(1+x^(2k))^3/(1+x^(6k))^2.
%F Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(3)) = 0.237425... . - _Amiram Eldar_, Jan 29 2024
%t f[p_, e_] := If[MemberQ[{1, 11}, Mod[p, 12]], (p^(e+1)-1)/(p-1), ((-p)^(e+1)-1)/(-p-1)]; f[2, e_] := 1; f[3, e_] := 1; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* _Amiram Eldar_, Sep 05 2023 *)
%o (PARI) a(n)=if(n<1, 0, direuler(p=2,n,1/(1-X)/(1-p*kronecker(12,p)*X))[n])
%o (PARI) {a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( 1-eta(x+A)*eta(x^3+A)*eta(x^4+A)^2*eta(x^6+A)^2/eta(x^12+A)^2, n))}
%o (PARI) my(N=99, q='x+O('x^N)); Vec(1-eta(q)*eta(q^3)*(eta(q^4)*eta(q^6))^2/eta(q^12)^2) \\ _Joerg Arndt_, Sep 05 2023
%Y Cf. A109039(n)=-a(n), if n>0.
%K sign,easy,mult
%O 1,5
%A _Michael Somos_, Jun 17 2005
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