OFFSET
-1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of 3 * (b(q)^2 / b(q^2)) / (c(q^2)^2 / c(q)) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of q^-1 * (phi(-q)^3 / phi(-q^3)) / ( psi(q^3)^3 / psi(q)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q) / eta(q^6))^5 * eta(q^3) / eta(q^2) in powers of q.
Euler transform of period 6 sequence [ -5, -4, -6, -4, -5, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (u + 8) * (v + 9) - (u - v)^2.
G.f.: (1/x) * (Product_{k>0} (1 + x^k) * (1 + x^(3*k)) * ((1 - x^(6*k)) / (1 - x^k))^4)^-1.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 72 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128638.
a(n) = A007258(n) = A045488(n) = A105559(n) = A128633(n) unless n = 0. Convolution inverse of A128638.
a(n) = -(-1)^n * A187146(n). - Michael Somos, May 20 2015
EXAMPLE
G.f. = 1/q - 5 + 6*q + 4*q^2 - 3*q^3 - 12*q^4 - 8*q^5 + 12*q^6 + 30*q^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 4 EllipticTheta[ 4, 0, q]^3 EllipticTheta[ 2, 0, q^(1/2)] / (EllipticTheta[ 4, 0, q^3] EllipticTheta[ 2, 0, q(3/2)]^3), {q, 0, n}]; (* Michael Somos, May 20 2015 *)
QP = QPochhammer; s = (QP[q]/QP[q^6])^5*(QP[q^3]/QP[q^2]) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from PARI *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^6 + A))^5 * eta(x^3 + A) / eta(x^2 + A), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Mar 15 2007
STATUS
approved