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A252706 Expansion of phi(-q) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

%I A252706 #39 Feb 16 2025 08:33:24
%S A252706 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 A252706 -58,0,74,-84,0,104,-116,0,144,-160,0,198,-220,0,268,-296,0,360,-396,
%U A252706 0,480,-528,0,634,-694,0,832,-908,0,1084,-1184,0,1404,-1528,0
%N A252706 Expansion of phi(-q) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.
%C A252706 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A252706 G. C. Greubel, <a href="/A252706/b252706.txt">Table of n, a(n) for n = 0..1000</a>
%H A252706 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A252706 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A252706 Expansion of f(-q, -q^2) / f(q, q^2) in powers of q where f(,) is Ramanujan's two-variable theta function.
%F A252706 Euler transform of period 6 sequence [ -2, -1, 0, -1, -2, 0, ...].
%F A252706 G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 3^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A101195.
%F A252706 G.f.: Product_{k>0} (1 - x^k + x^(2*k)) / (1 + x^k + x^(2*k)).
%F A252706 a(n) = (-1)^n * A139137(n).
%F A252706 Convolution inverse is A098151.
%F A252706 a(3*n + 2) = 0. a(3*n) = A098151(n). a(3*n + 1) = -2 * A097197(n).
%e A252706 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 A252706 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^3], {q, 0, n}];
%o A252706 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A)^2), n))};
%Y A252706 Cf. A097197, A098151, A101195, A139137.
%K A252706 sign
%O A252706 0,2
%A A252706 _Michael Somos_, Apr 04 2015