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A131124
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Expansion of q^(-1) * (phi(-q) / psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.
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3
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1, -4, 4, 0, 2, 0, -8, 0, -1, 0, 20, 0, -2, 0, -40, 0, 3, 0, 72, 0, 2, 0, -128, 0, -4, 0, 220, 0, -4, 0, -360, 0, 5, 0, 576, 0, 8, 0, -904, 0, -8, 0, 1384, 0, -10, 0, -2088, 0, 11, 0, 3108, 0, 12, 0, -4552, 0, -15, 0, 6592, 0, -18, 0, -9448, 0, 22, 0, 13392
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OFFSET
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-1,2
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COMMENTS
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Number 3 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(8). [Yang 2004] - Michael Somos, Jul 21 2014
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LINKS
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FORMULA
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Expansion of (eta(q)^2 * eta(q^4) / (eta(q^2) * eta(q^8)^2 ))^2 in powers of q.
Euler transform of period 8 sequence [ -4, -2, -4, -4, -4, -2, -4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A107035.
G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = u * (u + 8) * (v + 4) - v^2.
G.f.: (x * Product_{k>0} (1 + x^k)^4 * (1 + x^(2*k))^2 * (1 + x^(4*k))^4 )^-1.
a(2*n) = 0 unless n=0. a(n) = A029841(n) unless n=0. a(4*n - 1) = A029839(n). a(4*n + 1) = 4 * A079006(n).
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EXAMPLE
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G.f. = 1/q - 4 + 4*q + 2*q^3 - 8*q^5 - q^7 + 20*q^9 - 2*q^11 - 40*q^13 + 3*q^15 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 4 (EllipticTheta[ 4, 0, q] / EllipticTheta[ 2, 0, q^2])^2, {q, 0, n}]; (* Michael Somos, Apr 24 2015 *)
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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)^2 * eta(x^4 + A) / (eta(x^2 + A) * eta(x^8 + A)^2) )^2, 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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