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A014969
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Expansion of (theta_3(q) / theta_4(q))^2 in powers of q.
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17
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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
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OFFSET
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0,2
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COMMENTS
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REFERENCES
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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).
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LINKS
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FORMULA
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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(q)) = 1 / k'(q) in powers of the 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.
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
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = (1/4)*sqrt(8 + 6*sqrt(2)). - Simon Plouffe, Mar 02 2021
G.f.: A(q) = sqrt(-lambda(-q)/lambda(q)), where lambda(q) = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + ... is the elliptic modular function in powers of the nome q = exp(i*Pi*t), the g.f. of A115977; lambda(q) = k(q)^2, where k(q) = (theta_2(q) / theta_3(q))^2 is the elliptic modulus.
A(q) = sqrt(G(q)), where G(q) = 1 + 16q + 128*q^2 + 704*q^3 + 3072*q^4 + ... is the g.f. of A014972. (End)
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EXAMPLE
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G.f. = 1 + 8*q + 32*q^2 + 96*q^3 + 256*q^4 + 624*q^5 + 1408*q^6 + 3008*q^7 + ...
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MATHEMATICA
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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-François Alcover, Nov 09 2015, adapted from PARI *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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