OFFSET
0,2
COMMENTS
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^(-5/4) (eta(q) * eta(q^4)^4 / eta(q^2))^2 in powers of q.
Expansion of chi(-x)^2 * f(-x^4)^8 = psi(-x)^8 / chi(-x)^6 in powers of x where psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [ -2, 0, -2, -8, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1024 (t / i)^6 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A215600.
G.f.: Product_{k>0} (1 - x^(4*k))^8 * (1 - x^(2*k - 1))^2.
EXAMPLE
1 - 2*x + x^2 - 2*x^3 - 4*x^4 + 12*x^5 - 3*x^6 + 10*x^7 - 3*x^8 - 20*x^9 + ...
q^5 - 2*q^9 + q^13 - 2*q^17 - 4*q^21 + 12*q^25 - 3*q^29 + 10*q^33 - 3*q^37 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ q^4]^8 QPochhammer[ q, q^2]^2, {q, 0, n}]
a[ n_] := SeriesCoefficient[ (1/ 16) EllipticTheta[ 2, Pi/4, q^(1/2)]^8 / QPochhammer[ q, q^2]^6, {q, 0, n + 1}]
a[ n_] := SeriesCoefficient[ (1/2) QPochhammer[ q^4]^6 EllipticTheta[ 2, Pi/4, q^(1/2)]^2, {q, 0, n + 1/4}]
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^4 / eta(x^2 + A))^2, n))}
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, May 17 2013
STATUS
approved