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%I #19 Mar 12 2021 22:24:48
%S 1,-3,4,-5,8,-14,20,-25,37,-54,71,-91,121,-164,210,-264,343,-443,554,
%T -687,863,-1087,1340,-1637,2021,-2489,3027,-3659,4442,-5391,6480,
%U -7755,9306,-11153,13278,-15752,18711,-22203,26214,-30860,36354,-42777,50137,-58628
%N Expansion of f(-x, -x^5)^2 / f(x, x^3) in powers of x where f(, ) is Ramanujan's general 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="/A263041/b263041.txt">Table of n, a(n) for n = 0..1000</a>
%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)^3 * psi(x^3)^2 / f(-x^2)^4 in powers of x where psi(), f() are Ramanujan theta functions.
%F Expansion of q^(-13/24) * eta(q)^3 * eta(q^6)^4 / (eta(q^2)^4 * eta(q^3)^2) in powers of q.
%F Euler transform of period 6 sequence [ -3, 1, -1, 1, -3, -1, ...].
%F a(n) = - A053269(3*n + 2).
%F a(n) ~ (-1)^n * exp(sqrt(n/2)*Pi) / (6*sqrt(n)). - _Vaclav Kotesovec_, Apr 17 2016
%e G.f. = 1 - 3*x + 4*x^2 - 5*x^3 + 8*x^4 - 14*x^5 + 20*x^6 - 25*x^7 + 37*x^8 + ...
%e G.f. = q^13 - 3*q^37 + 4*q^61 - 5*q^85 + 8*q^109 - 14*q^133 + 20*q^157 - 25*q^181 + ...
%t a[ n_] := SeriesCoefficient[ x^(-1/2) QPochhammer[ x] (EllipticTheta[ 2, 0, x^(3/2)] / EllipticTheta[ 2, 0, x^(1/2)])^2, {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^6 + A)^4 / (eta(x^2 + A)^4 * eta(x^3 + A)^2), n))};
%Y Cf. A053269.
%K sign
%O 0,2
%A _Michael Somos_, Apr 17 2016
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