OFFSET
-1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1) * (phi(-q^4)^2 / (phi(q) * psi(q^4)))^2 in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q)^2 * eta(q^4)^7 / (eta(q^2)^5 * eta(q^8)^4))^2 in powers of q.
Euler transform of period 8 sequence [ -4, 6, -4, -8, -4, 6, -4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/8) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A232772.
EXAMPLE
G.f. = 1/q - 4 + 12*q - 32*q^2 + 66*q^3 - 128*q^4 + 232*q^5 - 384*q^6 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (2 EllipticTheta[ 4, 0, q^4]^2 / (EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^2]))^2, {q, 0, n}]
a[ n_] := SeriesCoefficient[ (2 EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2] / EllipticTheta[ 2, Pi/4, q]^2)^2, {q, 0, n}]
PROG
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A)^7 / (eta(x^2 + A)^5 * eta(x^8 + A)^4))^2, n))}
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Dec 10 2013
STATUS
approved