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Expansion of phi(x^6) * f(-x)^3 / f(-x^3) in powers of x where phi(), f() are Ramanujan theta functions.
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%I #9 Mar 12 2021 22:24:48

%S 1,-3,0,6,-3,0,1,-9,0,12,-3,0,6,-12,0,6,-3,0,7,-15,0,18,-6,0,0,-15,0,

%T 24,-6,0,6,-15,0,6,-9,0,7,-21,0,30,-3,0,6,-21,0,24,-6,0,12,-27,0,0,-9,

%U 0,12,-21,0,36,-6,0,1,-18,0,36,-12,0,6,-33,0,18,-9,0

%N Expansion of phi(x^6) * f(-x)^3 / f(-x^3) in powers of x where phi(), f() are Ramanujan theta functions.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H G. C. Greubel, <a href="/A259659/b259659.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 phi(x^6) * b(x) in powers of x where phi() is a Ramanujan theta function and b() is a cubic AGM theta function.

%F Expansion of q^(-3/4) * eta(q)^3 * eta(q^12)^2 / (eta(q^3) * eta(q^6)) in powers of q.

%F Euler transform of period 12 sequence [ -3, -3, -2, -3, -3, -1, -3, -3, -2, -3, -3, -3, ...].

%F a(2*n + 1) = -3 * A227595(n). a(3*n + 1) = -3 * A259655(n). a(3*n + 2) = 0.

%e G.f. = 1 - 3*x + 6*x^3 - 3*x^4 + x^6 - 9*x^7 + 12*x^9 - 3*x^10 + 6*x^12 + ...

%e G.f. = q^3 - 3*q^7 + 6*q^15 - 3*q^19 + q^27 - 9*q^31 + 12*q^39 - 3*q^43 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^6] QPochhammer[ x]^3 / QPochhammer[ x^3], {x, 0, n}];

%t eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-3/4)* eta[q]^3*eta[q^12]^2/(eta[q^3]*eta[q^6]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Mar 17 2018 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^12 + A)^2 / (eta(x^3 + A) * eta(x^6 + A)), n))};

%Y Cf. A227595, A259655.

%K sign

%O 0,2

%A _Michael Somos_, Jul 02 2015