A262968 Expansion of phi(-q^6) / phi(-q) in powers of q where phi() is a Ramanujan theta function.

1, 2, 4, 8, 14, 24, 38, 60, 92, 138, 204, 296, 424, 600, 840, 1164, 1598, 2176, 2940, 3944, 5256, 6960, 9164, 12000, 15634, 20270, 26160, 33616, 43020, 54840, 69648, 88140, 111164, 139748, 175136, 218832, 272646, 338760, 419792, 518880, 639780, 786976, 965820

Offset: 0

Author

Michael Somos, Oct 05 2015

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 2*q + 4*q^2 + 8*q^3 + 14*q^4 + 24*q^5 + 38*q^6 + 60*q^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A262967.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^6] / EllipticTheta[ 4, 0, q], {q, 0, n}];

PARI

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

Formula

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

Euler transform of period 12 sequence [ 2, 1, 2, 1, 2, -1, 2, 1, 2, 1, 2, 0, ...].

a(n) = A262967(3*n).

a(n) ~ 5^(1/4) * exp(sqrt(5*n/6)*Pi) / (2^(9/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2015