A132979 Expansion of psi(q^3) / psi(q)^3 in powers of q where psi() is a Ramanujan theta function.

1, -3, 6, -12, 24, -45, 78, -132, 222, -363, 576, -900, 1392, -2121, 3180, -4716, 6936, -10098, 14550, -20796, 29520, -41595, 58176, -80856, 111750, -153561, 209820, -285240, 385968, -519840, 696960, -930516, 1237470, -1639314, 2163456, -2845080, 3728904, -4871211

Offset: 0

Author

Michael Somos, Sep 07 2007

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - 3*q + 6*q^2 - 12*q^3 + 24*q^4 - 45*q^5 + 78*q^6 - 132*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. A107760, A132974.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ 4 EllipticTheta[ 2, 0, q^(3/2)] / EllipticTheta[ 2, 0, q^(1/2)]^3, {q, 0, n}]; (* Michael Somos, Jun 20 2015 *)

PARI

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

Formula

Expansion of eta(q)^3 * eta(q^6)^2 / ( eta(q^2)^6 * eta(q^3) ) in powers of q.

Euler transform of period 6 sequence [ -3, 3, -2, 3, -3, 2, ...].

G.f.: Product_{k>0} (1 + x^(3*k)) * (1 - x^(6*k)) / ( (1 + x^k) * (1 - x^(2*k)) )^3.

a(n) = (-1)^n * A132974(n). Convolution inverse of A107760.