A128692 Expansion of (theta_4(q) / theta_3(q))^4 in powers of q.

1, -16, 128, -704, 3072, -11488, 38400, -117632, 335872, -904784, 2320128, -5702208, 13504512, -30952544, 68901888, -149403264, 316342272, -655445792, 1331327616, -2655115712, 5206288384, -10049485312, 19115905536, -35867019904, 66437873664

Offset

0,2

Comments

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

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.

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

Eric Weisstein's World of Mathematics, Elliptic Lambda Function

Formula

Expansion of 1 - lambda(tau) = lambda( -1 / tau) in powers of q = exp(pi i tau).

Expansion of (eta(q^4) * eta(q)^2 / eta(q^2)^3)^8 in powers of q.

Expansion of (phi(-q) / phi(q))^4 = (phi(-q) / phi(-q^2))^8 = (phi(-q^2) / phi(q))^8 = (f(-q) / f(q))^8 = (chi(-q) / chi(q))^8 = (psi(-q) / psi(q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.

Euler transform of period 4 sequence [ -16, 8, -16, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v^2 * (1 - u)^2 - 16 * u * (1 - v).

G.f.: (Product_{k>0} (1 - x^(2*k - 1)) / (1 + x^(2*k - 1)))^8 = exp( -16 * Sum_{k>0} x^(2*k - 1) * sigma(2*k - 1) / (2*k - 1)).

A014972(n) = (-1)^n * a(n). Convolution inverse of A014972.

Empirical : Sum_{n=1..infinity} (exp(-2*Pi)^(n-1)*a(n)) = -16+12*2^(1/2). - Simon Plouffe, Feb. 20, 2011.

Empirical : Sum_{n=1..infinity} exp(-Pi*sqrt(3))^(n-1)*(-1)^(n+1)*a(n) = 8 - 4*sqrt(3). - Simon Plouffe, Feb. 20, 2011.

Example

1 - 16*q + 128*q^2 - 704*q^3 + 3072*q^4 - 11488*q^5 + 38400*q^6 + ...

Mathematica

CoefficientList[(QPochhammer[q]/QPochhammer[-q])^8 + O[q]^30, q] (* Jean-François Alcover, Nov 05 2015 *)
eta[q_] := q^(1/24)*QPochhammer[q]; a[n_] := SeriesCoefficient[(eta[q^4]* eta[q]^2/eta[q^2]^3)^8, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jan 18 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A) / eta(x^2 + A)^3)^8, n))}
(PARI) {a(n) = if( n<0, 0, polcoeff( exp(-16 * sum(k=1, (n+1)\2, sigma(2*k-1) / (2*k-1) * x^(2*k-1), x * O(x^n))), n))}

Crossrefs

Cf. A014972.

Sequence in context: A035473 A014972 A115977 * A132136 A163399 A067488

Adjacent sequences: A128689 A128690 A128691 * A128693 A128694 A128695

Keyword

sign

Author

Michael Somos, Mar 20 2007

Status

approved