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A014969 Expansion of (theta_3(q) / theta_4(q))^2 in powers of q. 17
1, 8, 32, 96, 256, 624, 1408, 3008, 6144, 12072, 22976, 42528, 76800, 135728, 235264, 400704, 671744, 1109904, 1809568, 2914272, 4640256, 7310592, 11404416, 17626944, 27009024, 41047992, 61905088, 92681664 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed, 1895, p. 380, Section 488.

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

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).

R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 375. Eqs. (17),(18),(19)

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 11.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (phi(q) / phi(-q))^2 = (phi(q) / phi(-q^2))^4 = (phi(-q^2) / phi(-q))^4 = (psi(q) / psi(-q))^4 = (chi(q)^2 / chi(-q^2))^4 = (chi(q) / chi(-q))^4 = (chi(-q^2) / chi(-q)^2)^4 = (f(q) / f(-q))^4 in powers of q where phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos, Aug 01 2011

Expansion of Fricke t(omega) = tau(omega) / 2 + 1 in powers of q = exp(2 Pi i omega).

Expansion of elliptic 1 / sqrt(1 - lambda(z)) = 1 / k' in powers of nome q = exp(Pi*i*z).

Euler transform of period 4 sequence [ 8, -4, 8, 0, ...]. - Michael Somos, Jul 07 2005

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + u)^2 - 4*u*v^2. - Michael Somos, Nov 14 2006

G.f.: (theta_3(x) / theta_4(x))^2 = (Sum_{k} x^k^2) / (Sum_{k} (-x)^k^2)^2 = (Product_{k>0} (1 - x^(4*k - 2)) / ((1 - x^(4*k - 1)) * (1 - x^(4*k - 3)))^2)^4.

A139820(n) = (-1)^n * a(n). 8 * A107035(n) = a(n) unless n=0. 2 * A131126(n) = a(n) unless n=0. Convolution inverse of A139820.

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A029841. - Michael Somos, Jun 04 2015

a(n) ~ exp(Pi*sqrt(2*n)) / (8 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 28 2015

G.f.: exp(8*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019

EXAMPLE

G.f. = 1 + 8*q + 32*q^2 + 96*q^3 + 256*q^4 + 624*q^5 + 1408*q^6 + 3008*q^7 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ 1 / Sqrt[1 - InverseEllipticNomeQ  @ q], {q, 0, n}]; (* Michael Somos, Aug 01 2011 *)

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q])^2, {q, 0, n}]; (* Michael Somos, Aug 01 2011 *)

nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^4, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)

s = (QPochhammer[q^2]^3/(QPochhammer[q]^2*QPochhammer[q^4]))^4+O[q]^30; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 09 2015, adapted from PARI *)

PROG

(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)))^4, n))}; /* Michael Somos, Jul 07 2005 */

CROSSREFS

Cf. A029841, A107035, A131126, A139820.

Sequence in context: A053348 A019256 A286399 * A139820 A241204 A195590

Adjacent sequences:  A014966 A014967 A014968 * A014970 A014971 A014972

KEYWORD

nonn,nice,changed

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 23 14:15 EDT 2019. Contains 322386 sequences. (Running on oeis4.)