OFFSET
-1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..1000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994).
FORMULA
Expansion of q^(1/4)*(eta(q^2)*eta(q^4)/(eta(q)*eta(q^8)))^2 in powers of q. -G. C. Greubel, Jun 23 2018
a(n) ~ exp(sqrt(n/2)*Pi) / (2^(7/4) * n^(3/4)). - Vaclav Kotesovec, Jun 28 2018
From Michael Somos, Jan 17 2023: (Start)
Expansion of (psi(x)/psi(-x^2))^2 = (phi(-x^4)/psi(-x))^2 = (chi(x)*chi(x^2))^2 = (chi(-x^4)/chi(-x))^2 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (128 t)) = f(t) where q = exp(2 Pi i t). (End)
EXAMPLE
T32b = 1/q + 2*q^3 + 3*q^7 + 6*q^11 + 7*q^15 + 10*q^19 + 16*q^23 + 20*q^27 + ...
MATHEMATICA
eta[q_] := q^(1/24)*QPochhammer[q]; a := CoefficientList[Series[q^(1/4)*(eta[q^2]*eta[q^4]/(eta[q]*eta[q^8]))^2, {q, 0, 60}], q]; Table[a[[n]], {n, 0, 50}] (* G. C. Greubel, Jun 23 2018 *)
From Michael Somos, Jan 17 2023: (Start)
a[ n_] := SeriesCoefficient[ (EllipticTheta[2, 0, q^(1/2)]/EllipticTheta[2, Pi/4, q])^2 / (2/q^(1/4)), {q, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[3, 0, -q^4]/EllipticTheta[2, Pi/4, q^(1/2)])^2 *(2*q^(1/4)), {q, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[-q, q^2]*QPochhammer[-q^2, q^4])^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[+q^4, q^8]/QPochhammer[+q, q^2])^2, {q, 0, n}]; (End)
PROG
(PARI) q='q+O('q^50); Vec((eta(q^2)*eta(q^4)/(eta(q)*eta(q^8)))^2) \\ G. C. Greubel, Jun 23 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 27 2000
EXTENSIONS
Terms a(6) onward added by G. C. Greubel, Jun 23 2018
STATUS
approved