%I #23 Mar 12 2021 22:24:48
%S 1,2,3,6,10,16,25,38,55,80,115,160,223,306,415,560,747,988,1301,1700,
%T 2206,2850,3661,4676,5950,7536,9500,11936,14936,18620,23141,28662,
%U 35386,43566,53480,65466,79937,97356,118277,143370,173391,209232,251966,302806
%N Expansion of psi(x^4) / chi(-x)^2 in powers of x where psi(), chi() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A260599/b260599.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)^4 / (f(-x)^2 * phi(x^2)) in powers of x where phi(), f() are Ramanujan theta functions.
%F Expansion of q^(-7/12) * eta(q^2)^2 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)) in powers of q.
%F Euler transform of period 8 sequence [ 2, 0, 2, 1, 2, 0, 2, -1, ...].
%F 2 * a(n) = A260574(4*n + 2).
%F a(n) ~ exp(sqrt(2*n/3)*Pi) / (8*sqrt(2*n)). - _Vaclav Kotesovec_, Oct 14 2015
%e G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 25*x^6 + 38*x^7 + ...
%e G.f. = q^7 + 2*q^19 + 3*q^31 + 6*q^43 + 10*q^55 + 16*q^67 + 25*q^79 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^2 EllipticTheta[ 2, 0, x^2] / (2 x^(1/2)), {x, 0, n}];
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^4]^4 / (QPochhammer[ x]^2 EllipticTheta[ 3, 0, x^2]), {x, 0, n}];
%t nmax=60; CoefficientList[Series[Product[(1+x^k)^2 * (1-x^(8*k))^2 / (1-x^(4*k)),{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)^2 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)), n))};
%Y Cf. A260574.
%K nonn
%O 0,2
%A _Michael Somos_, Jul 29 2015
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