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A112128 Expansion of phi(q^4) / phi(q) in powers of q where phi() is a Ramanujan theta function.

%I A112128 #20 Feb 16 2025 08:32:58
%S A112128 1,-2,4,-8,16,-28,48,-80,128,-202,312,-472,704,-1036,1504,-2160,3072,
%T A112128 -4324,6036,-8360,11488,-15680,21264,-28656,38400,-51182,67864,-89552,
%U A112128 117632,-153836,200352,-259904,335872,-432480,554952,-709728,904784,-1149916,1457136
%N A112128 Expansion of phi(q^4) / phi(q) in powers of q where phi() is a Ramanujan theta function.
%C A112128 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A112128 G. C. Greubel, <a href="/A112128/b112128.txt">Table of n, a(n) for n = 0..1000</a>
%H A112128 C. Adiga and N. Anitha, <a href="http://dx.doi.org/10.1017/S0004972700034742">A note on a continued fraction of Ramanujan</a>, Bull. Austral. Math. Soc. 70 (2004), pp. 489-497. MR2103981 (2005g:11009)
%H A112128 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A112128 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A112128 Expansion of (eta(q) / eta(q^16))^2 * (eta(q^8) / eta(q^2))^5 in powers of q.
%F A112128 Euler transform of period 16 sequence [ -2, 3, -2, 3, -2, 3, -2, -2, -2, 3, -2, 3, -2, 3, -2, 0, ...].
%F A112128 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - (1 - 2*u + 2*u^2) * (1 - 2*v + 2*v^2).
%F A112128 G.f.: (Sum_{k in Z} x^(4*k^2)) / (Sum_{k in Z} x^(k^2)) = theta_3(0, x^4) / theta_3(0, x).
%F A112128 G.f.: Product_{k>0} ((1 + x^(2*k)) * (1 + x^(4*k)))^3 / ((1 + x^k) * (1 + x^(8*k)))^2.
%F A112128 Expansion of continued fraction 1 / (1 + 2*x / (1 - x^2 + (x^1 + x^3)^2 / (1 - x^6 + (x^2 + x^6)^2 / (1 - x^10 + (x^3 + x^9)^2 / ...)))).
%F A112128 G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 1/2 * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A208724.
%F A112128 (-1)^n * a(n) = A208933(n). a(2*n) = A131126(n). a(2*n + 2) = -2 * A093160(n). - _Michael Somos_, Dec 11 2016
%F A112128 Convolution inverse of A208274. - _Michael Somos_, Dec 11 2016
%F A112128 a(n) ~ (-1)^n * exp(sqrt(n)*Pi) / (2^(7/2) * n^(3/4)). - _Vaclav Kotesovec_, Nov 15 2017
%e A112128 G.f. = 1 - 2*q + 4*q^2 - 8*q^3 + 16*q^4 - 28*q^5 + 48*q^6 - 80*q^7 + 128*q^8 + ...
%t A112128 QP = QPochhammer; s = QP[q]^2*(QP[q^8]^5/QP[q^2]^5/QP[q^16]^2) + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 30 2015, adapted from PARI *)
%t A112128 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^4] / EllipticTheta[ 3, 0, q], {q, 0, n}]; (* _Michael Somos_, Dec 11 2016 *)
%o A112128 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^8 + A)^5 / (eta(x^2 + A)^5 * eta(x^16 + A)^2), n))};
%Y A112128 Cf. A093160, A131126, A208724, A208933.
%K A112128 sign
%O A112128 0,2
%A A112128 _Michael Somos_, Aug 27 2005