OFFSET
-1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (eta(q) * eta(q^4) * eta(q^10) / (eta(q^2) * eta(q^5) * eta(q^20)))^2 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 5 / f(t) where q = exp(2 Pi i t).
Expansion of q^(-1) * (psi(-q) / psi(-q^5))^2 in powers of q where psi() is a Ramanujan theta function.
a(2*n) = -2 * A138522(n).
Convolution square of A145708.
a(n) = -(-1)^n * A138516(n). - Michael Somos, Sep 04 2015
EXAMPLE
G.f. = 1/q - 2 + q - 2*q^2 + 2*q^3 + 2*q^4 - q^5 - 4*q^7 + 2*q^8 + 5*q^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, Pi/4, q^(1/2)] / EllipticTheta[ 2, Pi/4, q^(5/2)])^2, {q, 0, n}]; (* Michael Somos, Sep 04 2015 *)
QP = QPochhammer; s = (QP[q]*QP[q^4]*(QP[q^10]/(QP[q^2]*QP[q^5]*QP[q^20]) ))^2 + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from PARI *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^10 + A) / (eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A)))^2, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 17 2008
STATUS
approved