OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(1/2) * eta(q^2)^5 / (eta(q)^2 * eta(q^4) * eta(q^8)^2) in powers of q.
Given g.f. A(x), then B(q) = (A(q^2) / q)^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^2 - (v - 4) * (u - 4)^2.
Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^3 - v) * (v^3 + u) - 3*u*v * (2*(u^2 + v^2) - 11). - Michael Somos, Jul 05 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A208850. - Michael Somos, Jul 05 2014
EXAMPLE
G.f. = 1 + 2*x + x^4 - 2*x^5 - x^8 + 4*x^9 - 6*x^13 + x^16 + 8*x^17 - 12*x^21 + ...
G.f. = 1/q + 2*q + q^7 - 2*q^9 - q^15 + 4*q^17 - 6*q^25 + q^31 + 8*q^33 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 2 q^(1/2) EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q^2], {q, 0, n}]; (* Michael Somos, Jul 05 2014 *)
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) * eta(x^8 + A)^2), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Feb 29 2012
STATUS
approved