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%I #23 Mar 12 2021 22:24:46
%S 1,2,6,12,22,40,68,112,182,286,440,668,996,1464,2128,3056,4342,6116,
%T 8538,11820,16248,22176,30068,40528,54308,72378,95976,126648,166352,
%U 217560,283344,367552,474998,611624,784812,1003712,1279562,1626216,2060708,2603856
%N Expansion of phi(q^2) / phi(-q) in powers of q where phi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A208850/b208850.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of eta(q^4)^5 / (eta(q)^2 * eta(q^2) * eta(q^8)^2) in powers of q.
%F Euler transform of period 8 sequence [ 2, 3, 2, -2, 2, 3, 2, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A208589.
%F G.f.: (Sum_k x^(2 * k^2)) / (Sum_k (-1)^k * x^k^2).
%F a(n) ~ exp(sqrt(n)*Pi)/(8*n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015
%e 1 + 2*q + 6*q^2 + 12*q^3 + 22*q^4 + 40*q^5 + 68*q^6 + 112*q^7 + 182*q^8 + ...
%t nmax=60; CoefficientList[Series[Product[(1-x^(4*k))^5 / ((1-x^k)^2 * (1-x^(2*k)) * (1-x^(8*k))^2),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 13 2015 *)
%t a[n_] := SeriesCoefficient[EllipticTheta[3, 0, q^2]/EllipticTheta[3, 0, -q], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Nov 27 2017 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^5 / (eta(x + A)^2 * eta(x^2 + A) * eta(x^8 + A)^2), n))}
%Y Cf. A208589.
%K nonn
%O 0,2
%A _Michael Somos_, Mar 02 2012
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