OFFSET
0,4
COMMENTS
REFERENCES
B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 354 Eq. (3.11)
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 chi(x) / chi(x^3)^3 in powers of x where chi() is a Ramanujan theta function.
Expansion of -q^(-1/3) * c(q^2) / c(-q) in powers of q where c() is a cubic AGM theta function.
Expansion of q^(-1/3) * (eta(q^2)^2 * eta(q^3)^3 * eta(q^12)^3) / (eta(q) * eta(q^4) * eta(q^6)^6) in powers of q.
Given g.f. A(x), then B(q) = q*A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (1 - u^3) * (1 - v^3) - (1 + 2*u*v) * (1 - u*v)^3.
Given g.f. A(x), then B(q) = q*A(q^3) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (v + v^2 + v^3) - u^3*(1 - 2*v + 4*v^2).
Given g.f. A(x), then B(q) = q*A(q^3) satisfies 0 = f(B(q), B(q^7)) where f(u, v) = (u-v)^8 - u*(1 - u^3) * (1 + 8*u^3) * v * (1 - v^3) * (1 + 8*v^3).
Given g.f. A(x), then B(q) = q*A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = (v^2 - u) * (v^2 - w) - v^2*(v - u^2) * (v - w^2).
Given g.f. A(x), then B(q) = q*A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u6 + u3 - u1*u2 + u6*u3 * (1 - 2*u1*u2).
Euler transform of period 12 sequence [ 1, -1, -2, 0, 1, 2, 1, 0, -2, -1, 1, 0, ...].
(-1)^n * a(n) = A092848(n).
Convolution inverse of A062244.
EXAMPLE
G.f. = 1 + x - 2*x^3 - 2*x^4 + x^5 + 4*x^6 + 4*x^7 - x^8 - 8*x^9 - 8*x^10 + ...
G.f. = q + q^4 - 2*q^10 - 2*q^13 + q^16 + 4*q^19 + 4*q^22 - q^25 - 8*q^28 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / QPochhammer[ -x^3, x^6]^3, {x, 0, n}]; (* Michael Somos, Feb 19 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x] / EllipticTheta[ 3, 0, x^9] - 1) / (2 x), {x, 0, 3 n}]; (* Michael Somos, Feb 19 2015 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^12 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^6), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Feb 14 2007
STATUS
approved