A192329 Expansion of q^(-1/3) * (eta(q) * eta(q^9))^2 / eta(q^3)^4 in powers of q.

1, -2, -1, 6, -7, -2, 20, -24, -8, 56, -61, -18, 137, -150, -46, 312, -327, -94, 663, -690, -199, 1342, -1366, -384, 2603, -2632, -739, 4884, -4869, -1344, 8890, -8808, -2422, 15784, -15487, -4212, 27389, -26728, -7242, 46608, -45155, -12136, 77888, -75102

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Formula

Expansion of q^(-1/3) * ( (b(q) * c(q^3)) / (c(q) * b(q^3)) )^(1/2) in powers of q where b(), c() are cubic AGM functions.

Euler transform of period 9 sequence [ -2, -2, 2, -2, -2, 2, -2, -2, 0, ...].

Given g.f. A(x) then B(x) = x * A(x^3) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = (u^2 - v) * (u - v^2) + 4*u^2*v^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (81 t)) = f(t) where q = exp(2 Pi i t).

G.f.: Product_{k>0} ((1 - x^k) * (1 - x^(9*k)))^2 / (1 - x^(3*k))^4.

Convolution inverse of A058601.

Example

G.f. = 1 - 2*x - x^2 + 6*x^3 - 7*x^4 - 2*x^5 + 20*x^6 - 24*x^7 - 8*x^8 + ...
G.f. = q - 2*q^4 - q^7 + 6*q^10 - 7*q^13 - 2*q^16 + 20*q^19 - 24*q^22 - 8*q^25 + ...

Mathematica

QP = QPochhammer; s = (QP[q]*QP[q^9])^2/QP[q^3]^4 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^9 + A))^2 / eta(x^3 + A)^4, n))}

Crossrefs

Cf. A058601.

Sequence in context: A325139 A084312 A066752 * A059364 A258870 A217922

Adjacent sequences: A192326 A192327 A192328 * A192330 A192331 A192332

Keyword

sign

Author

Michael Somos, Jul 01 2011

Status

approved