A132966 Expansion of f(-x) * chi(x^2)^2 in powers of x where f(), chi() are Ramanujan theta functions.

1, -1, 1, -2, -1, 0, 1, 1, 2, -1, 0, -1, 0, 1, -1, 1, 2, -2, 1, -2, -3, 0, 0, 1, 2, 0, 1, -2, -2, 2, 0, 2, 3, -3, 1, -3, -3, 2, 0, 4, 4, -2, 0, -3, -3, 2, -2, 3, 5, -3, 1, -6, -6, 2, 0, 5, 6, -3, 1, -4, -6, 4, -2, 6, 7, -5, 3, -8, -9, 5, -1, 7, 9, -5, 2, -8

Offset

0,4

Comments

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

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

Formula

Expansion of q^(1/8) * eta(q^4)^4 * eta(q) / (eta(q^2)^2 * eta(q^8)^2) in powers of q.

Euler transform of period 8 sequence [ -1, 1, -1, -3, -1, 1, -1, -1, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 32^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132965.

G.f.: Product_{k>0} (1 - x^k) * (1 + x^(2*k))^2 / (1 + x^(4*k))^2.

Example

G.f. = 1 - x + x^2 - 2*x^3 - x^4 + x^6 + x^7 + 2*x^8 - x^9 - x^11 + x^13 + ...
G.f. = q^-1 - q^7 + q^15 - 2*q^23 - q^31 + q^47 + q^55 + 2*q^63 - q^71 - ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ -x^2, x^4]^2, {x, 0, n}]; (* Michael Somos, Oct 31 2015 *)

Prog

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

Crossrefs

Cf. A132965.

Sequence in context: A143110 A319662 A109294 * A037897 A190248 A054070

Adjacent sequences: A132963 A132964 A132965 * A132967 A132968 A132969

Keyword

sign

Author

Michael Somos, Aug 23 2007

Status

approved