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A327959 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. 0
1, -248, 4124, -34752, 213126, -1057504, 4530744, -17333248, 60655377, -197230000, 603096260, -1749556736, 4848776870, -12908659008, 33161242504, -82505707520, 199429765972, -469556091240, 1079330385764, -2426800117504, 5346409013164, -11558035326944 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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.
REFERENCES
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2. See page 392.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
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).
a(n) = (-1)^n * A007245(n).
EXAMPLE
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.
MATHEMATICA
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;
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( (-x * ellj( -x + x^2 * O(x^n)))^(1/3), n))};
CROSSREFS
Cf. A007245.
Sequence in context: A027654 A003916 A007245 * A178967 A030062 A030650
KEYWORD
sign
AUTHOR
Michael Somos, Sep 30 2019
STATUS
approved

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Last modified February 24 16:16 EST 2024. Contains 370307 sequences. (Running on oeis4.)