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A208589
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Expansion of phi(x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.
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3
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1, 2, 0, 0, 1, -2, 0, 0, -1, 4, 0, 0, 0, -6, 0, 0, 1, 8, 0, 0, 0, -12, 0, 0, -1, 18, 0, 0, -1, -24, 0, 0, 2, 32, 0, 0, 1, -44, 0, 0, -2, 58, 0, 0, -1, -76, 0, 0, 2, 100, 0, 0, 1, -128, 0, 0, -3, 164, 0, 0, -1, -210, 0, 0, 4, 264, 0, 0, 2, -332, 0, 0, -5, 416
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of q^(1/2) * eta(q^2)^5 / (eta(q)^2 * eta(q^4) * eta(q^8)^2) in powers of q.
Given g.f. A(x), then B(q) = (A(q^2) / q)^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^2 - (v - 4) * (u - 4)^2.
Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^3 - v) * (v^3 + u) - 3*u*v * (2*(u^2 + v^2) - 11). - Michael Somos, Jul 05 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A208850. - Michael Somos, Jul 05 2014
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A029838(n). a(4*n + 1) = 2 * A083365(n).
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EXAMPLE
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G.f. = 1 + 2*x + x^4 - 2*x^5 - x^8 + 4*x^9 - 6*x^13 + x^16 + 8*x^17 - 12*x^21 + ...
G.f. = 1/q + 2*q + q^7 - 2*q^9 - q^15 + 4*q^17 - 6*q^25 + q^31 + 8*q^33 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 2 q^(1/2) EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q^2], {q, 0, n}]; (* Michael Somos, Jul 05 2014 *)
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PROG
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(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)^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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