OFFSET
0,9
COMMENTS
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 11th equation.
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 f(x, x^5) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.
Expansion of q^(-5/24) * eta(q^3) * eta(q^12) / (eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, 0, 0, ...].
G.f.: (1 + x + x^5 + x^8 + x^16 + x^21 + ...) / (1 + x + x^3 + x^6 + x^10 + ...). [Ramanujan]
G.f.: 1 - x^3 * (1 - x) / (1 - x^4) + x^8 * (1 - x) * (1 - x^3) / ((1 - x^4) * (1 - x^8)) - ... [Ramanujan]
EXAMPLE
G.f. = 1 - x^3 + x^4 - x^7 + 2*x^8 - x^9 - 2*x^11 + 3*x^12 - x^13 - 3*x^15 + ...
G.f. = q^5 - q^77 + q^101 - q^173 + 2*q^197 - q^221 - 2*q^269 + 3*q^293 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {x}, {-x^2}, x^2, x^3], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/8) EllipticTheta[ 2, Pi/4, x^(3/2)] / QPochhammer[ x^4], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ x^(-5/24) (EllipticTheta[ 3, 0, x^(1/3)] - EllipticTheta[ 3, 0, x^3]) / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}]; (* Michael Somos, Jan 10 2017 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^12 + A) / (eta(x^4 + A) * eta(x^6 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jul 21 2008
STATUS
approved