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Expansion of phi(-q^3) / phi(-q^2) in powers of q where phi() is a Ramanujan theta function.
2

%I #14 Mar 12 2021 22:24:48

%S 1,0,2,-2,4,-4,8,-8,14,-16,24,-28,42,-48,68,-80,108,-128,170,-200,260,

%T -308,392,-464,584,-688,856,-1010,1240,-1460,1780,-2088,2526,-2960,

%U 3552,-4152,4956,-5776,6856,-7976,9416,-10928,12848,-14872,17412,-20116,23456

%N Expansion of phi(-q^3) / phi(-q^2) 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="/A262966/b262966.txt">Table of n, a(n) for n = 0..2500</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], Sep 30 2015

%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^3)^2 * eta(q^4) / (eta(q^2)^2 * eta(q^6)) in powers of q.

%F Euler transform of period 12 sequence [0, 2, -2, 1, 0, 1, 0, 1, -2, 2, 0, 0, ...].

%F a(n) ~ (-1)^n * exp(sqrt(n/2)*Pi) / (2^(9/4) * sqrt(3) * n^(3/4)). - _Vaclav Kotesovec_, Oct 06 2015

%e G.f. = 1 + 2*q^2 - 2*q^3 + 4*q^4 - 4*q^5 + 8*q^6 - 8*q^7 + 14*q^8 - 16*q^9 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3] / EllipticTheta[ 4, 0, q^2], {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A) / (eta(x^2 + A)^2 * eta(x^6 + A)), n))};

%o (PARI) q='q+O('q^99); Vec(eta(q^3)^2*eta(q^4)/(eta(q^2)^2*eta(q^6))) \\ _Altug Alkan_, Jul 31 2018

%K sign

%O 0,3

%A _Michael Somos_, Oct 05 2015