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A189925 Expansion of theta_4/theta_3 in powers of q.

%I A189925 #37 Feb 16 2025 08:33:14
%S A189925 1,-4,8,-16,32,-56,96,-160,256,-404,624,-944,1408,-2072,3008,-4320,
%T A189925 6144,-8648,12072,-16720,22976,-31360,42528,-57312,76800,-102364,
%U A189925 135728,-179104,235264,-307672,400704,-519808,671744,-864960,1109904
%N A189925 Expansion of theta_4/theta_3 in powers of q.
%C A189925 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A189925 In Baker [1890] page 94 is equation (1): sqrt(cos theta) = [[...]] = 1 - 4q + 8q^2 -[[...]] where cos theta = k'. - _Michael Somos_, Dec 31 2023
%D A189925 Arthur L. Baker, Elliptic Functions, John Wiley & Sons, NY, 1890.
%H A189925 G. C. Greubel, <a href="/A189925/b189925.txt">Table of n, a(n) for n = 0..1000</a>
%H A189925 G. Berger, <a href="https://doi.org/10.1090/S0002-9939-00-05476-9">Relations between cusp forms on congruence and noncongruence groups</a>, Proc. of the Amer. Math. Soc., vol. 128, (2000), 2869-2874. see p. 2870 equation (4).
%H A189925 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A189925 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A189925 Expansion of eta(q)^4 * eta(q^4)^2 / eta(q^2)^6 in powers of q.
%F A189925 Expansion of Jacobian elliptic function sqrt(k') in powers of q.
%F A189925 Expansion of phi(-q) / phi(q) = chi(-q)^2 / chi(q)^2 = psi(-q)^2 / psi(q)^2 = phi(-q)^2 / phi(-q^2)^2 = phi(-q^2)^2 / phi(q)^2 = chi(-q)^4 / chi(-q^2)^2 = chi(-q^2)^2 / chi(q)^4 = f(-q)^2 / f(q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
%F A189925 Euler transform of period 4 sequence [ -4, 2, -4, 0, ...].
%F A189925 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v^2 * (u^2 + 1) - 2*u.
%F A189925 Unique solution to f(x^2)^(-2) = (f(x) + 1/f(x)) / 2 and f(0) = 1, f'(0) nonzero.
%F A189925 G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A079006.
%F A189925 G.f.: theta_4 / theta_3 = (Sum_{k} (-x)^k^2)/(Sum_{k} x^k^2) = (Product_{k>0} ((1 - x^(4*k - 1)) * (1 - x^(4*k - 3)))^2 / (1 - x^(4*k - 2)))^2.
%F A189925 Convolution inverse of A007096. a(n) = (-1)^n * A007096(n). a(2*n) = A014969(n). a(2*n + 1) = -4 * A093160(n). a(4*n) = A097243(n). a(4*n + 2) = 8*A022577(n).
%F A189925 a(n) ~ (-1)^n * exp(Pi*sqrt(n))/(2^(5/2)*n^(3/4)). - _Vaclav Kotesovec_, Jul 04 2016
%F A189925 G.f.: exp(-4*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - _Ilya Gutkovskiy_, Apr 19 2019
%e A189925 G.f. = 1 - 4*q + 8*q^2 - 16*q^3 + 32*q^4 - 56*q^5 + 96*q^6 - 160*q^7 + 256*q^8 + ...
%t A189925 nmax = 50; CoefficientList[Series[Product[(1+x^(2*k))^2 / (1+x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 04 2016 *)
%t A189925 With[{nmax = 50}, CoefficientList[Series[4 QPochhammer[-1, x^2]^2/QPochhammer[-1, x]^4, {x, 0, nmax}], x]] (* _Jan Mangaldan_, Jan 04 2017 *)
%t A189925 With[{nmax = 50}, CoefficientList[Series[EllipticTheta[4, 0, x]/EllipticTheta[3, 0, x], {x, 0, nmax}], x]] (* _Jan Mangaldan_, Jan 04 2017 *)
%t A189925 a[ n_] :=  SeriesCoefficient[(1 - InverseEllipticNomeQ[x])^(1/4), {x, 0, n}]; (* _Michael Somos_, Dec 31 2023 *)
%o A189925 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A)^2 * eta(x^4 + A) / eta(x^2 + A)^3 )^2, n))};
%Y A189925 Cf. A007096, A014969, A079006, A093160.
%K A189925 sign
%O A189925 0,2
%A A189925 _Michael Somos_, May 01 2011