A199659 Expansion of q^(1/4) * (eta(q) / eta(q^3))^3 in powers of q.

1, -3, 0, 8, -9, 0, 17, -27, 0, 46, -57, 0, 98, -126, 0, 198, -243, 0, 371, -465, 0, 692, -828, 0, 1205, -1458, 0, 2082, -2463, 0, 3463, -4104, 0, 5678, -6642, 0, 9085, -10623, 0, 14370, -16632, 0, 22273, -25758, 0, 34178, -39246, 0, 51674, -59220, 0, 77362

Offset: 0

Author

Michael Somos, Nov 08 2011

Keywords

• sign

Comments

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

Examples

			1 - 3*x + 8*x^3 - 9*x^4 + 17*x^6 - 27*x^7 + 46*x^9 - 57*x^10 + 98*x^12 + ...
1/q - 3*q^3 + 8*q^11 - 9*q^15 + 17*q^23 - 27*q^27 + 46*q^35 - 57*q^39 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A007262, A030182, A137569.

Programs

Mathematica

QP = QPochhammer; s = (QP[q]/QP[q^3])^3 + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^3 + A))^3, n))}

Formula

Expansion of (f(-x) / f(-x^3))^3 in powers of x where f() is a Ramanujan theta function.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 3^(3/2) / f(t) where q = exp(2 Pi i t).

G.f.: (Product_{k>0} (1 - x^k) / (1 - x^(3*k)))^3.

Convolution cube of A137569. Convolution square is A007262. Convolution fourth power is A030182.

a(3*n + 2) = 0.

a(0) = 1, a(n) = -(3/n)*Sum_{k=1..n} A046913(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 28 2017