OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^10)^5 / (eta(q^2)^5 * eta(q^5)^2 * eta(q^20)^2) in powers of q.
Euler transform of period 20 sequence [ -2, 3, -2, 1, 0, 3, -2, 1, -2, 0, -2, 1, -2, 3, 0, 1, -2, 3, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 - 2*u^2 + 5*u^4) * (1 - 2*v^2 + 5*v^4) - 4*(u^2 + 2*u*v - v^2)^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (v^2 + 3*u*v - u^2) * (u^2 + v^2) - u*v * (1 + 5*u^2*v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 5^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A144377.
G.f.: Product_{k>0} P(10, x^k)^3 * P(5, x^k) / P(20, x^k)^2 where P(n, x) is the n-th cyclotomic polynomial.
EXAMPLE
G.f. = 1 - 2*q + 4*q^2 - 8*q^3 + 14*q^4 - 22*q^5 + 36*q^6 - 56*q^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^5] / EllipticTheta[ 3, 0, q], {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^10 + A)^5 / (eta(x^2 + A)^5 * eta(x^5 + A)^2 * eta(x^20 + A)^2), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Sep 06 2015
STATUS
approved