A208604 Expansion of phi(-q) / phi(q^4) in powers of q where phi() is a Ramanujan theta function.

1, -2, 0, 0, 0, 4, 0, 0, 0, -10, 0, 0, 0, 20, 0, 0, 0, -36, 0, 0, 0, 64, 0, 0, 0, -110, 0, 0, 0, 180, 0, 0, 0, -288, 0, 0, 0, 452, 0, 0, 0, -692, 0, 0, 0, 1044, 0, 0, 0, -1554, 0, 0, 0, 2276, 0, 0, 0, -3296, 0, 0, 0, 4724, 0, 0, 0, -6696, 0, 0, 0, 9408, 0, 0

Offset

0,2

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 eta(q)^2 * eta(q^4)^2 * eta(q^16)^2 / (eta(q^2) * eta(q^8)^5) in powers of q.

Euler transform of period 16 sequence [ -2, -1, -2, -3, -2, -1, -2, 2, -2, -1, -2, -3, -2, -1, -2, 0, ...].

G.f.: (Sum_{k in Z} (-1)^k * x^k^2) / (Sum_{k in Z} x^(4 * k^2)).

a(4*n) = 0 unless n=0. a(4*n + 2) = a(4*n + 3) = 0. a(4*n + 1) = -2 * A079006(n).

a(n) = (-1)^n * A208274(n). Convolution inverse of A208933. - Michael Somos, Dec 11 2016

G.f.: Product_{k>0} (1 + x^(8*k)) / ((1 + x^k)^2 * (1 + x^(2*k)) * (1 + x^(4*k))^3). - Michael Somos, Dec 11 2016

Example

G.f. = 1 - 2*q + 4*q^5 - 10*q^9 + 20*q^13 - 36*q^17 + 64*q^21 - 110*q^25 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] / EllipticTheta[ 3, 0, q^4], {q, 0, n}]; (* Michael Somos, Dec 11 2016 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^16 + A)^2 / (eta(x^2 + A) * eta(x^8 + A)^5), n))};

Crossrefs

Cf. A079006, A208274, A208933.

Sequence in context: A051516 A236799 A208274 * A127391 A262162 A246950

Adjacent sequences: A208601 A208602 A208603 * A208605 A208606 A208607

Keyword

sign

Author

Michael Somos, Feb 29 2012

Status

approved