A327959 - OEIS

%I #10 Apr 16 2022 16:49:43

%S 1,-248,4124,-34752,213126,-1057504,4530744,-17333248,60655377,

%T -197230000,603096260,-1749556736,4848776870,-12908659008,33161242504,

%U -82505707520,199429765972,-469556091240,1079330385764,-2426800117504,5346409013164,-11558035326944

%N 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.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C 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.

%D S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2. See page 392.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of chi(x)^8 - 256 * x / chi(x)^16 in powers of x where chi() is a Ramanujan theta function.

%F 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.

%F 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.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = f(t) where q = exp(2 Pi i t).

%F a(n) = (-1)^n * A007245(n).

%e G.f. = 1 - 248*x + 4124*x^2 - 34752*x^3 + 213126*x^4 - 1057504*x^5 + ...

%e G.f. = q^-1 - 248*q^2 + 4124*q^5 - 34752*q^8 + 213126*q^11 - 1057504*q^14 + ...

%e 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.

%t 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;

%o (PARI) {a(n) = if( n<0, 0, polcoeff( (-x * ellj( -x + x^2 * O(x^n)))^(1/3), n))};

%Y Cf. A007245.

%K sign

%O 0,2

%A _Michael Somos_, Sep 30 2019