A260944 Expansion of phi(-x^4) * psi(-x^6) / chi(-x^3) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

1, 0, 0, 1, -2, 0, 0, -2, 0, 1, 0, 0, 1, -2, 0, 1, 0, 0, 1, 0, 0, 1, -2, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, -2, 0, 0, 0, 0, 0, -2, 0, 2, -2, 0, 1, 0, 0, 0, -4, 0, 0, 0, 0, 1, 0, 0, 1, -2, 0, 1, 0, 0, 2, 0, 0, 0, -2, 0, 2, -2, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0

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 + x^3 - 2*x^4 - 2*x^7 + x^9 + x^12 - 2*x^13 + x^15 + x^18 + x^21 + ...
G.f. = q^7 + q^31 - 2*q^39 - 2*q^63 + q^79 + q^103 - 2*q^111 + q^127 + ...
		

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. A112607, A131962.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^4] EllipticTheta[ 2, Pi/4, x^3]   QPochhammer[ -x^3, x^3] / (2^(1/2) x^(3/4)), {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^6 + A)^2 * eta(x^24 + A) / (eta(x^3 + A) * eta(x^8 + A) * eta(x^12 + A)), n))};

Formula

Expansion of q^(-7/8) * eta(q^4)^2 * eta(q^6)^2 * eta(q^24) / (eta(q^3) * eta(q^8) * eta(q^12)) in powers of q.

Euler transform of period 24 sequence [ 0, 0, 1, -2, 0, -1, 0, -1, 1, 0, 0, -2, 0, 0, 1, -1, 0, -1, 0, -2, 1, 0, 0, -2, ...].

a(3*n) = A131962(n). a(3*n + 1) = -2 * A112607(n-1). a(3*n + 2) = 0.