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A109040
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Expansion of 1-eta(q)*eta(q^3)*(eta(q^4)*eta(q^6))^2/eta(q^12)^2 in powers of q.
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1
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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, 1, -6, -28, -4, -30, 1, 12, -16, 24, 1, 38, -18, 14, -4, -40, -6, -42, 12, -4, 24, 48, 1, 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
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OFFSET
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1,5
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REFERENCES
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Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.68).
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LINKS
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FORMULA
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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).
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.
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(3)) = 0.237425... . - Amiram Eldar, Jan 29 2024
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X)/(1-p*kronecker(12, p)*X))[n])
(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))}
(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
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CROSSREFS
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KEYWORD
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sign,easy,mult
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AUTHOR
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STATUS
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approved
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