OFFSET
-1,4
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 (3 * c(q^2)) / (c(q) * c(q^4)) in powers of q where c() is a cubic AGM function.
Expansion of eta(q) * eta(q^4) * eta(q^6)^3 / (eta(q^2) * eta(q^3)^3 * eta(q^12)^3) in powers of q.
Euler transform of period 12 sequence [ -1, 0, 2, -1, -1, 0, -1, -1, 2, 0, -1, 2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (3/4)^(1/2) (t/i)^(-1) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A132974.
a(3*n + 1) = 0. a(3*n) = - A132974(n).
Convolution inverse of A113427.
a(3*n - 1) = A263993(n). - Michael Somos, Oct 31 2015
EXAMPLE
G.f. = 1/q - 1 + 2*q^2 - 3*q^3 + 4*q^5 - 6*q^6 + 10*q^8 - 12*q^9 + 20*q^11 - ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 2 EllipticTheta[ 2, Pi/4, q^(1/2)] / EllipticTheta[ 2, Pi/4, q^(3/2)]^3, {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)
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^6 + A)^3 / (eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^12 + A)^3), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Sep 18 2007
STATUS
approved