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A259662 Expansion of phi(-q^3) / phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.

%I A259662 #17 Feb 16 2025 08:33:26
%S A259662 1,6,24,78,222,576,1392,3180,6936,14550,29520,58176,111750,209820,
%T A259662 385968,696960,1237470,2163456,3728904,6343068,10658880,17708412,
%U A259662 29108880,47373696,76378992,122058870,193435248,304134558,474609180,735374016,1131698448,1730375436
%N A259662 Expansion of phi(-q^3) / phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.
%C A259662 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A259662 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A259662 G. C. Greubel, <a href="/A259662/b259662.txt">Table of n, a(n) for n = 0..1000</a>
%H A259662 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 A259662 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A259662 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A259662 Expansion of 1 / (2*a(q^2) - a(q)) = b(q^2) / b(q)^2 in powers of q where a(), b() are cubic AGM theta functions.
%F A259662 Expansion of eta(q^2)^3 * eta(q^3)^2 / (eta(q)^6 * eta(q^6)) in powers of q.
%F A259662 Euler transform of period 6 sequence [ 6, 3, 4, 3, 6, 2, ...].
%F A259662 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w^2*(u + v)^2 - 2*u*v^2*(v+w).
%F A259662 G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 432^(-1/2) (t/I)^-1 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A258093.
%F A259662 G.f.: Product_{k>0} (1 + x^k)^3 * (1 - x^(3*k)) / ((1 + x^(3*k)) * (1 - x^k)^3).
%F A259662 a(n) = A132974(2*n) = A132979(2*n).
%F A259662 Convolution inverse of A122859.
%F A259662 a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(5/4) * n^(5/4)). - _Vaclav Kotesovec_, Oct 14 2015
%e A259662 G.f. = 1 + 6*x + 24*x^2 + 78*x^3 + 222*x^4 + 576*x^5 + 1392*x^6 + ...
%t A259662 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] / EllipticTheta[ 4, 0, x]^3, {x, 0, n}];
%t A259662 nmax=60; CoefficientList[Series[Product[(1+x^k)^3 * (1-x^(3*k)) / ((1+x^(3*k)) * (1-x^k)^3),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 14 2015 *)
%o A259662 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^2 / (eta(x + A)^6 * eta(x^6 + A)), n))};
%Y A259662 Cf. A122859, A132974, A132979, A258093.
%K A259662 nonn
%O A259662 0,2
%A A259662 _Michael Somos_, Jul 02 2015