OFFSET
0,5
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 q^(-1/2) * theta_2(q^3) / theta_2(q) in powers of q^2.
Expansion of q^(-1/4) * eta(q) * eta(q^6)^2 / (eta(q^2)^2 * eta(q^3)) in powers of q.
Given g.f. A(x), B(q) = A(q^4) * q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = u^3 - v + 3*u*v^2 - 3*u^2*v^3.
Given g.f. A(x), B(q) = (A(q^4) * q)^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v + v^2 + 3*u^2*v.
a(n) = (-1)^n * A036018(n).
G.f.: Product_{k>0} (1 + x^k + x^(2*k)) * (1 - x^k + x^(2*k))^2.
a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
EXAMPLE
G.f. = 1 - x + x^2 - x^3 + 2*x^4 - 3*x^5 + 3*x^6 - 4*x^7 + 6*x^8 + ...
G.f. = q - q^5 + q^9 - q^13 + 2*q^17 - 3*q^21 + 3*q^25 - 4*q^29 + 6*q^33 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ x^(-1/4) EllipticTheta[ 2, 0, x^(3/2)] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Dec 03 2004
STATUS
approved