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

%I A128692 #41 Jul 04 2025 17:24:16
%S A128692 1,-16,128,-704,3072,-11488,38400,-117632,335872,-904784,2320128,
%T A128692 -5702208,13504512,-30952544,68901888,-149403264,316342272,-655445792,
%U A128692 1331327616,-2655115712,5206288384,-10049485312,19115905536,-35867019904,66437873664
%N A128692 Expansion of (theta_4(q) / theta_3(q))^4 in powers of q.
%C A128692 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A128692 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
%H A128692 G. C. Greubel, <a href="/A128692/b128692.txt">Table of n, a(n) for n = 0..1000</a>
%H A128692 Yaacov Kopeliovich, <a href="https://arxiv.org/abs/0705.3914">Modular equations of order p and Theta functions</a>, arXiv:0705.3914 [math.CV], 2007, p. 9 (theorem 3.1).
%H A128692 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A128692 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A128692 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EllipticLambdaFunction.html">Elliptic Lambda Function</a>
%F A128692 Expansion of 1 - lambda(tau) = lambda( -1 / tau) in powers of q = exp(pi  i tau).
%F A128692 Expansion of (eta(q^4) * eta(q)^2 / eta(q^2)^3)^8 in powers of q.
%F A128692 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.
%F A128692 Euler transform of period 4 sequence [ -16, 8, -16, 0, ...].
%F A128692 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).
%F A128692 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)).
%F A128692 A014972(n) = (-1)^n * a(n). Convolution inverse of A014972.
%F A128692 Empirical : Sum_{n=1..infinity} (exp(-2*Pi)^(n-1)*a(n)) = -16+12*2^(1/2). - _Simon Plouffe_, Feb 20 2011.
%F A128692 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.
%F A128692 G.f. A(x) satisfies A(x) = 1/A(-x). - _Michael Somos_, Oct 07 2024
%F A128692 G.f. A(q) satisfies 1 = (A(q)*A(q^3))^(1/4) + ((1 - A(q))*(1 - A(q^3)))^(1/4). See Kopeliovich link. - _Paul D. Hanna_, Oct 14 2024
%e A128692 G.f. A(q) = 1 - 16*q + 128*q^2 - 704*q^3 + 3072*q^4 - 11488*q^5 + 38400*q^6 + ...
%e A128692 RELATED SERIES.
%e A128692 (A(q)*A(q^3))^(1/4) = 1 - 4*x + 8*x^2 - 20*x^3 + 48*x^4 - 88*x^5 + 168*x^6 - 320*x^7 + 544*x^8 - 932*x^9 + ... + (-1)^n*A123861(n)*x^n + ...
%e A128692 where ((1 - A(q))*(1 - A(q^3)))^(1/4) = 1 - (A(q)*A(q^3))^(1/4).
%t A128692 CoefficientList[(QPochhammer[q]/QPochhammer[-q])^8 + O[q]^30, q] (* _Jean-François Alcover_, Nov 05 2015 *)
%t A128692 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 *)
%o A128692 (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)^8, n))};
%o A128692 (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))};
%Y A128692 Cf. A014972, A123861.
%K A128692 sign
%O A128692 0,2
%A A128692 _Michael Somos_, Mar 20 2007