A143161 Expansion of chi(-x)^2 * chi(-x^2) in powers of x where chi() is a Ramanujan theta function.

1, -2, 0, 0, 3, -2, 0, 0, 4, -6, 0, 0, 7, -8, 0, 0, 13, -14, 0, 0, 19, -20, 0, 0, 29, -34, 0, 0, 43, -46, 0, 0, 62, -70, 0, 0, 90, -96, 0, 0, 126, -138, 0, 0, 174, -186, 0, 0, 239, -262, 0, 0, 325, -346, 0, 0, 435, -472, 0, 0, 580, -620, 0, 0, 769, -826, 0, 0

Offset: 0

Author

Michael Somos, Jul 27 2008

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - 2*x + 3*x^4 - 2*x^5 + 4*x^8 - 6*x^9 + 7*x^12 - 8*x^13 + 13*x^16 + ...
G.f. = 1/q - 2*q^5 + 3*q^23 - 2*q^29 + 4*q^47 - 6*q^53 + 7*q^71 - 8*q^77 + ...
		

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. A029552, A098613.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 QPochhammer[ x^2, x^4], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)

PARI

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

Formula

Expansion of q^(1/6) * eta(q)^2 / (eta(q^2) * eta(q^4)) in powers of q.

Euler transform of period 4 sequence [ -2, -1, -2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 8^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A098613. - Michael Somos, Sep 07 2015

a(4*n + 2) = a(4*n + 3) = 0.

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

a(4*n) = A029552(n). a(4*n + 1) = -2 * A098613(n).