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A131986
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Expansion of (eta(q) / eta(q^9))^3 in powers of q.
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4
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1, -3, 0, 5, 0, 0, -7, 0, 0, 3, 0, 0, 15, 0, 0, -32, 0, 0, 9, 0, 0, 58, 0, 0, -96, 0, 0, 22, 0, 0, 149, 0, 0, -253, 0, 0, 68, 0, 0, 372, 0, 0, -599, 0, 0, 140, 0, 0, 826, 0, 0, -1317, 0, 0, 317, 0, 0, 1768, 0, 0, -2735, 0, 0, 632, 0, 0, 3526, 0, 0, -5434, 0, 0
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OFFSET
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-1,2
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COMMENTS
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Number 4 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Jul 21 2014
A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(9). [Yang 2004] - Michael Somos, Jul 21 2014
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LINKS
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FORMULA
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Euler transform of period 9 sequence [ -3, -3, -3, -3, -3, -3, -3, -3, 0, ...].
G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = (u + v)^3 - u*v * (27 + 9*(u+v) + u*v).
G.f. A(q) satisfies 0 = f(A(q), A(q^2), A(q^4)) where f(u, v, w) = u^2 + w^2 + u*w - v^2*(u+w) - 6*v^2 - 6*v*(u+w) - 27*v.
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 27 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A121589.
a(3*n + 1) = 0. a(3*n) = 0 unless n=0. a(3*n - 1) = A058091(n).
G.f.: (1/x) * (Product_{k>0} (1 - x^k) / (1 - x^(9*k)))^3.
G.f. A(q) satisfies 0 = f(A(q), A(q^3)) where f(u, v) = (27 + 9*u + u^2) * (27 + 9*v + v^2) * u - v^3. - Michael Somos, May 13 2021
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EXAMPLE
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G.f. = 1/q - 3 + 5*q^2 - 7*q^5 + 3*q^8 + 15*q^11 - 32*q^14 + 9*q^17 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q] / QPochhammer[ q^9]))^3, {q, 0, n}]; (* Michael Somos, Jul 21 2014 *)
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PROG
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(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^9 + A))^3, n))};
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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