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

0,2

Comments

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

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

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.

Example

G.f. = 1 + 2*q + 2*q^3 + 6*q^4 + 6*q^6 + 16*q^7 + 14*q^9 + 36*q^10 + ...

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 *)

Prog

(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))};

Crossrefs

Cf. A139214, A139215.

Sequence in context: A208385 A186634 A363910 * A344873 A306079 A388481

Adjacent sequences: A139210 A139211 A139212 * A139214 A139215 A139216

Keyword

nonn

Author

Michael Somos, Apr 11 2008

Status

approved