A263398 Expansion of phi(-x^2)^6 * psi(x^6) / f(x)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

1, -2, -7, 14, 20, -36, -34, 40, 50, -30, -71, 76, 82, -144, -98, 112, 131, -70, -140, 170, 168, -288, -228, 232, 246, -120, -290, 258, 310, -468, -280, 344, 337, -190, -350, 394, 412, -648, -510, 496, 462, -252, -583, 558, 602, -864, -532, 584, 664, -350

Offset: 0

Author

Michael Somos, Oct 16 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 - 7*x^2 + 14*x^3 + 20*x^4 - 36*x^5 - 34*x^6 + 40*x^7 + ...
G.f. = q^2 - 2*q^5 - 7*q^8 + 14*q^11 + 20*q^14 - 36*q^17 - 34*q^20 + ...
		

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. A245643.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2]^6 EllipticTheta[ 2, 0, x^3] / (2 x^(3/4) QPochhammer[ -x]^2), {x, 0, n}];

PARI

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

PARI

q='q+O('q^99); Vec(eta(q)^2*eta(q^2)^6*eta(q^12)^2/(eta(q^4)^4*eta(q^6))) \\ Altug Alkan, Jul 31 2018

Formula

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

Euler transform of period 12 sequence [-2, -8, -2, -4, -2, -7, -2, -4, -2, -8, -2, -5, ...].

8 * a(n) = A245643(3*n + 2).