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%I #12 Mar 12 2021 22:24:45
%S 1,2,0,-2,-2,0,4,4,0,-6,-8,0,10,12,0,-16,-18,0,24,28,0,-36,-40,0,52,
%T 58,0,-74,-84,0,104,116,0,-144,-160,0,198,220,0,-268,-296,0,360,396,0,
%U -480,-528,0,634,694,0,-832,-908,0,1084,1184,0,-1404,-1528,0,1808,1964,0,-2316,-2514,0,2952,3196
%N Expansion of phi(q) / phi(q^3) in powers of q where phi() is a Ramanujan 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="/A139137/b139137.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 f(q, -q^2) / f(-q, q^2) in powers of q where f(,) is Ramanujan's two-variable theta function. - _Michael Somos_, Apr 04 2015
%F Expansion of eta(q^2)^5 * eta(q^3)^2 * eta(q^12)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^6)^5) in powers of q.
%F Euler transform of period 12 sequence [ 2, -3, 0, -1, 2, 0, 2, -1, 0, -3, 2, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132002. - _Michael Somos_, Apr 04 2015
%F G.f.: (Sum_{k in Z} x^k^2) / (Sum_{k in Z} x^(3*k^2)).
%F G.f.: Product_{k>0} P(12, x^k)^2 / (P(3, x^k) * P(6, x^k)^3) where P(n, x) is n-th cyclotomic polynomial.
%F Convolution inverse of A132002. - _Michael Somos_, Apr 04 2015
%F a(n) = (-1)^n * A252706(n). - _Michael Somos_, Apr 04 2015
%F a(3*n + 2) = 0. a(3*n) = A132002(n). a(3*n + 1) = 2 * A139135(n).
%e G.f. = 1 + 2*q - 2*q^3 - 2*q^4 + 4*q^6 + 4*q^7 - 6*q^9 - 8*q^10 + 10*q^12 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^3], {q, 0, n}]; (* _Michael Somos_, Apr 04 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^2 * eta(x^12 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^5), n))};
%Y Cf. A132002, A139135, A252706.
%K sign
%O 0,2
%A _Michael Somos_, Apr 10 2008
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