A260941 Expansion of phi(-x) * phi(x^6) / chi(-x^3) in powers of x where phi(), chi() are Ramanujan theta functions.

1, -2, 0, 1, 0, 0, 3, -4, 0, 2, -2, 0, 2, 0, 0, 1, -4, 0, 0, 0, 0, 3, 0, 0, 2, -4, 0, 4, -2, 0, 2, 0, 0, 0, -8, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 0, 0, 0, 2, -4, 0, 2, -6, 0, 2, 0, 0, 4, -4, 0, 0, -4, 0, 1, 0, 0, 4, 0, 0, 2, 0, 0, 2, 0, 0, 1, -4, 0, 0, -4

Offset: 0

Author

Michael Somos, Aug 04 2015

Keywords

• sign

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 - 2*x + x^3 + 3*x^6 - 4*x^7 + 2*x^9 - 2*x^10 + 2*x^12 + x^15 + ...
G.f. = q - 2*q^9 + q^25 + 3*q^49 - 4*q^57 + 2*q^73 - 2*q^81 + 2*q^97 + ...
		

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

Crossrefs

Cf. A112608, A131961.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 3, 0, x^6] QPochhammer[ -x^3, x^3], {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^12 + A)^5 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) * eta(x^24 + A)^2), n))};

PARI

q='q+O('q^99); Vec(eta(q)^2*eta(q^12)^5/(eta(q^2)*eta(q^3)*eta(q^6)*eta(q^24)^2)) \\ Altug Alkan, Aug 01 2018

Formula

Expansion of q^(-1/8) * eta(q)^2 * eta(q^12)^5 / (eta(q^2) * eta(q^3) * eta(q^6) * eta(q^24)^2) in powers of q.

Euler transform of period 24 sequence [ -2, -1, -1, -1, -2, 1, -2, -1, -1, -1, -2, -4, -2, -1, -1, -1, -2, 1, -2, -1, -1, -1, -2, -2, ...].

a(3*n) = A131961(n). a(3*n + 1) = -2 * A112608(n). a(3*n + 2) = 0.