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A208933 Expansion of phi(q^4) / phi(-q) in powers of q where phi() is a Ramanujan theta function.

%I A208933 #31 Feb 16 2025 08:33:16
%S A208933 1,2,4,8,16,28,48,80,128,202,312,472,704,1036,1504,2160,3072,4324,
%T A208933 6036,8360,11488,15680,21264,28656,38400,51182,67864,89552,117632,
%U A208933 153836,200352,259904,335872,432480,554952,709728,904784,1149916,1457136,1841200,2320128
%N A208933 Expansion of phi(q^4) / phi(-q) in powers of q where phi() is a Ramanujan theta function.
%C A208933 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A208933 G. C. Greubel, <a href="/A208933/b208933.txt">Table of n, a(n) for n = 0..1000</a>
%H A208933 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016.
%H A208933 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A208933 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A208933 Expansion of eta(q^2) * eta(q^8)^5 / (eta(q) * eta(q^4) * eta(q^16))^2 in powers of q.
%F A208933 Euler transform of period 16 sequence [ 2, 1, 2, 3, 2, 1, 2, -2, 2, 1, 2, 3, 2, 1, 2, 0, ...].
%F A208933 G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/4) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A208603.
%F A208933 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2*u - 1) * (2*v^2 - 2*v + 1) - u^2.
%F A208933 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = 4 * u * (u - 1) * (2*u - 1) * v * (v - 1) * (2*v - 1) - (u - v)^4.
%F A208933 (-1)^n * a(n) = A112128(n). a(n) = 2 * A123655(n) unless n=0. 2 * a(n) = A007096(n) unless n=0. a(2*n) = A131126(n). a(2*n + 1) = 2 * A093160(n). Convolution inverse of A208604.
%F A208933 G.f.: (Sum_{k in Z} x^(4 * k^2)) / (Sum_{k in Z} (-1)^k * x^(k^2)) = theta_3(x^4) / theta_3(-x).
%F A208933 G.f.: Product_{k>0} ((1 + x^(2*k)) * (1 + x^(4*k)))^3 / ((1 + (-x)^k) * (1 + x^(8*k)))^2.
%F A208933 a(n) ~ exp(sqrt(n)*Pi) / (2^(7/2) * n^(3/4)). - _Vaclav Kotesovec_, Oct 14 2015
%e A208933 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 A208933 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^4] / EllipticTheta[ 4, 0, q], {q, 0, n}]; (* _Michael Somos_, Apr 25 2015 *)
%t A208933 nmax=60; CoefficientList[Series[Product[(1-x^(2*k)) * (1-x^(8*k))^5 / ((1-x^k)^2 * (1-x^(4*k))^2 * (1-x^(16*k))^2),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 14 2015 *)
%o A208933 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^8 + A)^5 / (eta(x + A) * eta(x^4 + A) * eta(x^16 + A))^2, n))};
%Y A208933 Cf. A093160, A112128, A123655, A131126, A208603, A208604.
%K A208933 nonn
%O A208933 0,2
%A A208933 _Michael Somos_, Mar 13 2012