OFFSET
0,10
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 f(-x^2, -x^4) / f(x, x^5) in powers of x where f() is Ramanujan's two-variable theta function. - Michael Somos, Apr 05 2015
Expansion of q^(1/4) * eta(q) * eta(q^4) * eta(q^6) / ( eta(q^2) * eta(q^3) * eta(q^12) ) in powers of q.
Euler transform of period 12 sequence [ -1, 0, 0, -1, -1, 0, -1, -1, 0, 0, -1, 0, ...].
Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (1 + v^4) - (1 + u*v)^3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = 3^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A036018.
Convolution inverse is A036018. Convolution square is A062243. Convolution 4th power is A187147. - Michael Somos, Apr 05 2015
EXAMPLE
G.f. = 1 - x -x^4 + x^6 - x^7 + 2*x69 - x^10 + 2*x^12 - 2*x^13 + 2*x^15 + ...
G.f. = 1/q - q^3 - q^15 + q^23 - q^27 + 2*q^35 - q^39 + 2*q^47 - 2*q^51 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ q^(1/4) EllipticTheta[ 2, Pi/4, q^(1/2)] / EllipticTheta[ 2, Pi/4, q^(3/2)], {q, 0, n}]; (* Michael Somos, Apr 05 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Nov 22 2007, Nov 23 2007
STATUS
approved