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%I #9 Mar 12 2021 22:24:48
%S 1,2,4,10,20,36,64,112,189,308,492,778,1210,1844,2776,4144,6114,8914,
%T 12884,18484,26302,37124,52040,72512,100415,138196,189160,257648,
%U 349184,470932,632312,845472,1125853,1493222,1973060,2597892,3408754,4457600,5810544
%N Expansion of f(-x, x^2) / f(-x, -x^3)^3 in powers of x where f(, ) is Ramanujan's general 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="/A263993/b263993.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^3) / (phi(-x) * f(-x^4)^2) in powers of x where phi(), f() are Ramanujan theta functions.
%F Expansion of q^(1/3) * eta(q^2) * eta(q^6)^5 / (eta(q)^2 * eta(q^3)^2 * eta(q^4)^2 * eta(q^12)^2) in powers of q.
%F Euler transform of period 12 sequence [ 2, 1, 4, 3, 2, -2, 2, 3, 4, 1, 2, 2, ...].
%F a(n) = A133637(3*n - 1).
%e G.f. = 1 + 2*x + 4*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 64*x^6 + 112*x^7 + ...
%e G.f. = 1/q + 2*q^2 + 4*q^5 + 10*q^8 + 20*q^11 + 36*q^14 + 64*q^17 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3] / (EllipticTheta[ 4, 0, x] QPochhammer[ x^4]^2), {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^5 / (eta(x + A)^2 * eta(x^3 + A)^2 * eta(x^4 + A)^2 * eta(x^12 + A)^2), n))};
%Y Cf. A133637.
%K nonn
%O 0,2
%A _Michael Somos_, Oct 31 2015
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