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.

1, -248, 4124, -34752, 213126, -1057504, 4530744, -17333248, 60655377, -197230000, 603096260, -1749556736, 4848776870, -12908659008, 33161242504, -82505707520, 199429765972, -469556091240, 1079330385764, -2426800117504, 5346409013164, -11558035326944

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

Table of n, a(n) for n=0..21.

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

Adjacent sequences: A327956 A327957 A327958 * A327960 A327961 A327962

Keyword

sign

Author

Michael Somos, Sep 30 2019

Status

approved