A227175 Expansion of (phi(x) / f(-x^4))^4 in powers of x where phi(), f() are Ramanujan theta functions.

1, 8, 24, 32, 28, 80, 192, 192, 134, 408, 864, 800, 568, 1520, 3072, 2752, 1809, 4808, 9456, 8192, 5316, 13616, 26112, 22144, 13990, 35376, 66624, 55584, 34696, 86016, 159744, 131392, 80724, 198256, 363720, 295776, 180068, 436816, 793344, 638976, 384940

Offset

0,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(2/3) * (eta(q^2)^5 / (eta(q)^2 * eta(q^4)^3))^4 in powers of q.

Euler transform of period 4 sequence [ 8, -12, 8, 0, ...].

a(2*n + 1) = 8 * A022569(n). Convolution square of A227033.

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 8 * exp(-2*Pi/3) * sqrt(2) = A388803. - Simon Plouffe, Sep 18 2025

Example

1 + 8*x + 24*x^2 + 32*x^3 + 28*x^4 + 80*x^5 + 192*x^6 + 192*x^7 + 134*x^8 + ...
q^-2 + 8*q + 24*q^4 + 32*q^7 + 28*q^10 + 80*q^13 + 192*q^16 + 192*q^19 + ...

Mathematica

a[ n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]/QPochhammer[q^4])^4, {q, 0, n}];

Prog

(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)^3))^4, n))}

Crossrefs

Cf. A022569, A227033.

Sequence in context: A096727 A028660 A028644 * A340930 A269420 A056196

Adjacent sequences: A227172 A227173 A227174 * A227176 A227177 A227178

Keyword

nonn

Author

Michael Somos, Jul 03 2013

Status

approved