A014972 Expansion of (theta_3(q) / theta_4(q) )^4 in powers of q; also of 1 / (1 - lambda(z)).

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

The relation with A092877 is equivalent to eta(q^2)^24 = eta(q)^16 * eta(q^4)^8 + 16 * eta(q)^8 * eta(q^4)^16. - Michael Somos, Apr 11 2004

References

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Elliptic Lambda Function

Formula

Expansion of 1 / (1 - lambda(t)) = 1 / lambda(-1 / t) in powers of q = exp(Pi i t).

Expansion of (phi(q) / phi(-q))^4 = (phi(-q^2) / phi(-q))^8 = (phi(q) / phi(-q^2))^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.

Expansion of (eta(q^2)^3 / (eta(q^4) * eta(q)^2))^8 in powers of q. - Michael Somos, Apr 11 2004

Euler transform of period 4 sequence [ 16, -8, 16, 0, ...]. - Michael Somos, Apr 11 2004

G.f. A(x) satisfies A(-x) = 1 / A(x). Also 0 = f(A(x), A(x^2)) where f(u, v) = (u - 1)^2 + 16 * u*v * (1 - v). - Michael Somos, Apr 11 2004

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)). - Michael Somos, Apr 11 2004

a(n) = 16 * A092877(n) unless n = 0. a(n) = A132136(n) unless n = 0. Convolution inverse of A128692.

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

a(n) ~ exp(2*Pi*sqrt(n)) / (32 * n^(3/4)). - Vaclav Kotesovec, Aug 28 2015

Example

G.f. = 1 + 16*q + 128*q^2 + 704*q^3 + 3072*q^4 + 11488*q^5 + 38400*q^6 + 117632*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q])^4, {q, 0, n}]; (* Michael Somos, Jun 25 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 / (QPochhammer[ q^4] QPochhammer[ q]^2))^8, {q, 0, n}]; (* Michael Somos, Jun 25 2014 *)
nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^8, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)

Prog

(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))}; /* Michael Somos, Apr 11 2004 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)))^8, n))}; /* Michael Somos, Apr 11 2004 */

Crossrefs

Cf. A005798, A007248, A029845, A092877, A128692, A132136.

Sequence in context: A008416 A045651 A035473 * A115977 A128692 A132136

Adjacent sequences: A014969 A014970 A014971 * A014973 A014974 A014975

Keyword

nonn

Author

N. J. A. Sloane

Status

approved