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A233673 Expansion of phi(q) * phi(q^9) / phi(q^3)^2 in powers of q where phi() is a Ramanujan theta function.

%I A233673 #18 Feb 16 2025 08:33:21
%S A233673 1,2,0,-4,-6,0,12,16,0,-28,-36,0,60,76,0,-120,-150,0,228,280,0,-416,
%T A233673 -504,0,732,878,0,-1252,-1488,0,2088,2464,0,-3408,-3996,0,5460,6364,0,
%U A233673 -8600,-9972,0,13344,15400,0,-20424,-23472,0,30876,35346,0,-46152,-52644
%N A233673 Expansion of phi(q) * phi(q^9) / phi(q^3)^2 in powers of q where phi() is a Ramanujan theta function.
%C A233673 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A233673 G. C. Greubel, <a href="/A233673/b233673.txt">Table of n, a(n) for n = 0..2500</a>
%H A233673 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A233673 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A233673 Expansion of eta(q^2)^5 * eta(q^3)^4 * eta(q^12)^4 * eta(q^18)^5 / (eta(q)^2 * eta(q^4)^2 * eta(q^6)^10 * eta(q^9)^2 * eta(q^36)^2) in powers of q.
%F A233673 Euler transform of period 36 sequence [ 2, -3, -2, -1, 2, 3, 2, -1, 0, -3, 2, 1, 2, -3, -2, -1, 2, 0, 2, -1, -2, -3, 2, 1, 2, -3, 0, -1, 2, 3, 2, -1, -2, -3, 2, 0, ...].
%F A233673 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = f(t) where q = exp(2 Pi i t).
%F A233673 a(n) = 2 * A233670(n) unless n=0.
%e A233673 G.f. = 1 + 2*q - 4*q^3 - 6*q^4 + 12*q^6 + 16*q^7 - 28*q^9 - 36*q^10 + ...
%t A233673 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^9] / EllipticTheta[ 3, 0, q^3]^2, {q, 0, n}]; (* _Michael Somos_, Aug 27 2015 *)
%o A233673 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^4 * eta(x^12 + A)^4 * eta(x^18 + A)^5 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^10 * eta(x^9 + A)^2 * eta(x^36 + A)^2), n))};
%Y A233673 Cf. A233670.
%K A233673 sign
%O A233673 0,2
%A A233673 _Michael Somos_, Dec 14 2013