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A328789 Expansion of (chi(x^3) / chi(-x^2))^2 in powers of x where chi() is a Ramanujan theta function.

%I A328789 #9 Feb 16 2025 08:33:58
%S A328789 1,0,2,2,3,4,7,6,11,14,17,22,32,34,49,60,72,90,117,132,171,206,245,
%T A328789 298,369,422,522,620,728,868,1043,1198,1439,1688,1962,2304,2717,3114,
%U A328789 3668,4258,4909,5698,6627,7566,8788,10112,11574,13310,15317,17410,20010
%N A328789 Expansion of (chi(x^3) / chi(-x^2))^2 in powers of x where chi() is a Ramanujan theta function.
%C A328789 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A328789 Convolution square of A097242.
%C A328789 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328795.
%H A328789 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A328789 Expansion of q^(1/12) * (eta(q^4) * eta(q^6)^2)^2 / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.
%F A328789 Euler transform of period 12 sequence [0, 2, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, ...].
%F A328789 G.f.: Product_{k>=1} (1 + x^(6*k - 3))^2 / (1 - x^(4*k - 2))^2.
%F A328789 a(n) = A112206(2*n).
%F A328789 a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Oct 31 2019
%e A328789 G.f. = 1 + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 6*x^7 + 11*x^8 + ...
%e A328789 G.f. = q^-1 + 2*q^23 + 2*q^35 + 3*q^47 + 4*q^59 + 7*q^71 + 6*q^83 + ...
%t A328789 a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^2, x^2] QPochhammer[ -x^3, x^6])^2, {x, 0, n}];
%o A328789 (PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A) * eta(x^6 + A)^2)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))};
%Y A328789 Cf. A097242, A112206, A328795.
%K A328789 nonn
%O A328789 0,3
%A A328789 _Michael Somos_, Oct 27 2019