A139213 Expansion of phi(q) * phi(-q^18) / (phi(-q^3) * phi(-q^6)) in powers of q where phi() is a Ramanujan theta function.

1, 2, 0, 2, 6, 0, 6, 16, 0, 14, 36, 0, 30, 76, 0, 60, 150, 0, 114, 280, 0, 208, 504, 0, 366, 878, 0, 626, 1488, 0, 1044, 2464, 0, 1704, 3996, 0, 2730, 6364, 0, 4300, 9972, 0, 6672, 15400, 0, 10212, 23472, 0, 15438, 35346, 0, 23076, 52644, 0, 34134, 77616, 0

Offset: 0

Author

Michael Somos, Apr 11 2008

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 + 2*q^3 + 6*q^4 + 6*q^6 + 16*q^7 + 14*q^9 + 36*q^10 + ...
		

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. A139214, A139215.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^18] / (EllipticTheta[ 4, 0, q^3] EllipticTheta[ 4, 0, q^6]), {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^12 + A) * eta(x^18 + A)^2 / (eta(x + A)^2 * eta(x^3 + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^36 + A)), n))};

Formula

Expansion of eta(q^2)^5 * eta(q^12) * eta(q^18)^2 / (eta(q)^2 * eta(q^3)^2 * eta(q^4)^2 * eta(q^6) * eta(q^36)) in powers of q.

Euler transform of period 36 sequence [ 2, -3, 4, -1, 2, 0, 2, -1, 4, -3, 2, 1, 2, -3, 4, -1, 2, -2, 2, -1, 4, -3, 2, 1, 2, -3, 4, -1, 2, 0, 2, -1, 4, -3, 2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139215.

a(n) = 2 * A139214(n) unless n=0. a(3*n + 2) = 0.