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A327959
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Expansion of (-j(1/2 + t))^(1/3) * q^(1/3) in powers of q = exp(2 Pi i t) where j is the modular j-function.
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0
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1, -248, 4124, -34752, 213126, -1057504, 4530744, -17333248, 60655377, -197230000, 603096260, -1749556736, 4848776870, -12908659008, 33161242504, -82505707520, 199429765972, -469556091240, 1079330385764, -2426800117504, 5346409013164, -11558035326944
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OFFSET
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0,2
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COMMENTS
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Given g.f. A(x), then B(q) = A(q^3) / q satisfies J_n = B(sqrt(-n)/2)/32 where a few values of J_n as given in Ramanujan, Notebooks, Vol. 2, page 392.
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REFERENCES
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S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2. See page 392.
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LINKS
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FORMULA
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Expansion of chi(x)^8 - 256 * x / chi(x)^16 in powers of x where chi() is a Ramanujan theta function.
Expansion of (phi(x)^8 - (2 * phi(x) * phi(-x))^4 + 16 * phi(-x)^8) / f(x)^8 in powers of x where phi(), f() are Ramanujan theta functions.
Expansion of q^(1/3) * (eta(q)^2 / (eta(q) * eta(q^4)))^8 + 256 * (eta(q) * eta(q^4) / eta(q^2))^16 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = f(t) where q = exp(2 Pi i t).
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EXAMPLE
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G.f. = 1 - 248*x + 4124*x^2 - 34752*x^3 + 213126*x^4 - 1057504*x^5 + ...
G.f. = q^-1 - 248*q^2 + 4124*q^5 - 34752*q^8 + 213126*q^11 - 1057504*q^14 + ...
If J_n := (-j(1/2 + sqrt(-n)/2))^(1/3) / 32, then J_3 = 0, J_11 = 1, J_19 = 3, J_43 = 30, J_67 = 165, J_163 = 20010.
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ With[ {m = InverseEllipticNomeQ[q]}, (1 - 16 m (1 - m)) / (4 m (1 - m))^(1/3)] 4 (-q)^(1/3), {q, 0, n}] // Simplify;
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( (-x * ellj( -x + x^2 * O(x^n)))^(1/3), 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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