A261326 Expansion of f(-x^2, -x^4)^2 / (f(x^3, -x^6) * f(-x, x^2)) in powers of x where f(,) is Ramanujan's general theta function.

1, 1, -2, -4, -3, 4, 12, 8, -10, -28, -18, 24, 60, 38, -48, -120, -75, 92, 228, 140, -172, -416, -252, 304, 732, 439, -524, -1252, -744, 884, 2088, 1232, -1450, -3408, -1998, 2336, 5460, 3182, -3704, -8600, -4986, 5772, 13344, 7700, -8872, -20424, -11736

Offset: 0

Author

Michael Somos, Aug 14 2015

Keywords

• sign

Comments

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

Examples

			G.f. = 1 + x - 2*x^2 - 4*x^3 - 3*x^4 + 4*x^5 + 12*x^6 + 8*x^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. A261320, A261325.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ x^2] QPochhammer[ x^6] / QPochhammer[ -x^3]^3, {x, 0, n}];

PARI

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

Formula

Expansion of f(x) * f(-x^2) * f(-x^6) / f(x^3)^3 in powers of x where f() is a Ramanujan theta function.

Euler transform of period 12 sequence [ 1, -3, -2, -2, 1, 2, 1, -2, -2, -3, 1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261325.

a(3*n) = A261320(n). a(3*n + 1) = A261325(n).