A233670 Expansion of q * phi(-q^2) * psi(q^9) / (f(q^3) * phi(q^3)) in powers of q where f(), phi(), psi() are Ramanujan theta functions.

1, 0, -2, -3, 0, 6, 8, 0, -14, -18, 0, 30, 38, 0, -60, -75, 0, 114, 140, 0, -208, -252, 0, 366, 439, 0, -626, -744, 0, 1044, 1232, 0, -1704, -1998, 0, 2730, 3182, 0, -4300, -4986, 0, 6672, 7700, 0, -10212, -11736, 0, 15438, 17673, 0, -23076, -26322, 0, 34134

Offset: 1

Author

Michael Somos, Dec 14 2013

Keywords

• sign

Comments

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

Examples

			G.f. = q - 2*q^3 - 3*q^4 + 6*q^6 + 8*q^7 - 14*q^9 - 18*q^10 + 30*q^12 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A058647, A128638, A228447.

Programs

Mathematica

A233670[n_] := SeriesCoefficient[q*QPochhammer[q^2]^2*QPochhammer[q^3]^3 *QPochhammer[q^12]^3*QPochhammer[q^18]^2/(QPochhammer[q^4] * QPochhammer[q^6]^8*QPochhammer[q^9]), {q, 0, n}]; Table[A233670[n], {n, 50}] (* G. C. Greubel, Oct 09 2017 *)

PARI

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^12 + A)^3 * eta(x^18 + A)^2 / (eta(x^4 + A) * eta(x^6 + A)^8 * eta(x^9 + A)), n))};

Formula

Expansion of eta(q^2)^2 * eta(q^3)^3 * eta(q^12)^3 * eta(q^18)^2 / (eta(q^4) * eta(q^6)^8 * eta(q^9)) in powers of q.

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

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

a(3*n) = -2 * A228447(n). a(6*n) = 6 * A128638(n). A(3*n + 2) = 0.

Convolution inverse of A058647.