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%I #22 Mar 12 2021 22:24:48
%S 1,4,6,8,16,24,32,48,66,92,128,168,224,296,384,496,640,816,1030,1304,
%T 1632,2032,2528,3120,3840,4716,5760,7008,8512,10296,12416,14944,17922,
%U 21440,25600,30480,36208,42936,50784,59952,70656,83088,97536,114312,133728
%N Expansion of phi(x)^2 / phi(-x^2) in powers of x where phi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C G.f. is a period 1 Fourier series which satisfies f(-1/ (32 t)) = 8^(1/2) (t/i)^(1/2) g(t) where q = exp(2*Pi i t) and g() is the g.f. for A260313.
%H G. C. Greubel, <a href="/A260314/b260314.txt">Table of n, a(n) for n = 0..1000</a>
%H Vaclav Kotesovec, <a href="https://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 f(-x^4) * chi(x)^4 = psi(x) * chi(x)^3 = phi(-x^2)^3 / phi(-x)^2 = psi(x)^4 / f(-x^4)^3 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
%F Expansion of eta(q^2)^8 / (eta(q)^4 * eta(q^4)^3) in powers of q.
%F Euler transform of period 4 sequence [4, -4, 4, -1, ... ].
%F a(n) ~ exp(Pi*sqrt(n/2)) / (2*sqrt(2*n)). - _Vaclav Kotesovec_, Oct 14 2015
%e G.f. = 1 + 4*x + 6*x^2 + 8*x^3 + 16*x^4 + 24*x^5 + 32*x^6 + 48*x^7 + ...
%t a[ n_]:= SeriesCoefficient[ EllipticTheta[ 3, 0, x]^2 / EllipticTheta[ 4, 0, x^2], {x, 0, n}];
%t nmax=60; CoefficientList[Series[Product[(1-x^(2*k))^8 / ((1-x^k)^4 * (1-x^(4*k))^3),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 14 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 / (eta(x + A)^4 * eta(x^4 + A)^3), n))};
%o (PARI) q='q+O('q^99); Vec(eta(q^2)^8/(eta(q)^4*eta(q^4)^3)) \\ _Altug Alkan_, Mar 19 2018
%Y Cf. A260313.
%K nonn
%O 0,2
%A _Michael Somos_, Jul 22 2015
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