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A128128 Expansion of chi(-q^3) / chi^3(-q) in powers of q where chi() is a Ramanujan theta function.

%I A128128 #23 Jul 19 2025 13:06:56
%S A128128 1,3,6,12,21,36,60,96,150,228,342,504,732,1050,1488,2088,2901,3996,
%T A128128 5460,7404,9972,13344,17748,23472,30876,40413,52644,68268,88152,
%U A128128 113364,145224,185352,235734,298800,377514,475488,597108,747690,933672,1162824
%N A128128 Expansion of chi(-q^3) / chi^3(-q) in powers of q where chi() is a Ramanujan theta function.
%C A128128 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A128128 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A128128 G. C. Greubel, <a href="/A128128/b128128.txt">Table of n, a(n) for n = 0..1000</a>
%H A128128 Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, <a href="https://arxiv.org/abs/2507.10965">Convolutive sequences, I: Through the lens of integer partition functions</a>, arXiv:2507.10965 [math.CO], 2025. See pp. 4, 11, 13.
%H A128128 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 A128128 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A128128 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A128128 Expansion of eta(q^2)^3 * eta(q^3) / (eta(q)^3 * eta(q^6)) in powers of q.
%F A128128 Euler transform of period 6 sequence [ 3, 0, 2, 0, 3, 0, ...].
%F A128128 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 + v - 2*u*v^2.
%F A128128 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u + u^2 + u^3) - v^3*(1 - 2*u + 4*u^2).
%F A128128 G.f. A(x) satisfies 0 = f(A(x), A(x^5)) where f(u, v) = u^6 + v^6 - 16*u^5*v^5 + 20*u^4*v^4 + 10*u^2*v^2*(u^3 + v^3) - 20*u^3*v^3 - 5*u*v*(u^3 + v^3) + 5*u^2*v^2 - u*v.
%F A128128 Expansion of b(q^2) / b(q) in powers of q where b() is a cubic AGM theta function.
%F A128128 G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A062242.
%F A128128 a(n) = 3*A128129(n) unless n=0.
%F A128128 Convolution inverse of A141094. - _Michael Somos_, Feb 19 2015
%F A128128 a(n) ~ exp(2*sqrt(2*n)*Pi/3) / (2^(7/4) * sqrt(3) * n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015
%e A128128 G.f. = 1 + 3*q + 6*q^2 + 12*q^3 + 21*q^4 + 36*q^5 + 60*q^6 + 96*q^7 + ...
%t A128128 a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^3 QPochhammer[ q^3] / (QPochhammer[ q]^3 QPochhammer[ q^6]), {q, 0, n}]; (* _Michael Somos_, Feb 19 2015 *)
%t A128128 nmax=60; CoefficientList[Series[Product[(1-x^(2*k))^3 * (1-x^(3*k)) / ((1-x^k)^3 * (1-x^(6*k))),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 13 2015 *)
%o A128128 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) / (eta(x + A)^3 * eta(x^6 + A)), n))};
%Y A128128 Cf. A062242, A128129, A141094.
%K A128128 nonn
%O A128128 0,2
%A A128128 _Michael Somos_, Feb 15 2007