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A210030 Expansion of phi(-q) / phi(q^2) in powers of q where phi() is a Ramanujan theta function.

%I A210030 #15 Feb 16 2025 08:33:17
%S A210030 1,-2,-2,4,6,-8,-12,16,22,-30,-40,52,68,-88,-112,144,182,-228,-286,
%T A210030 356,440,-544,-668,816,996,-1210,-1464,1768,2128,-2552,-3056,3648,
%U A210030 4342,-5160,-6116,7232,8538,-10056,-11820,13872,16248,-18996,-22176,25844,30068
%N A210030 Expansion of phi(-q) / phi(q^2) in powers of q where phi() is a Ramanujan theta function.
%C A210030 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A210030 G. C. Greubel, <a href="/A210030/b210030.txt">Table of n, a(n) for n = 0..1000</a>
%H A210030 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A210030 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A210030 Expansion of eta(q)^2 * eta(q^2) * eta(q^8)^2 / eta(q^4)^5 in powers of q.
%F A210030 Euler transform of period 8 sequence [ -2, -3, -2, 2, -2, -3, -2, 0, ...].
%F A210030 G.f.: (Sum_k (-1)^k * x^k^2) / (Sum_k x^(2 * k^2)).
%F A210030 a(n) = (-1)^n * A080015(n) = (-1)^[(n + 1) / 4] * A080054(n).
%F A210030 Convolution inverse of A208850.
%e A210030 1 - 2*q - 2*q^2 + 4*q^3 + 6*q^4 - 8*q^5 - 12*q^6 + 16*q^7 + 22*q^8 + ...
%t A210030 a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, -q]/EllipticTheta[3, 0, q^2], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Dec 17 2017 *)
%o A210030 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A) * eta(x^8 + A)^2 / eta(x^4 + A)^5, n))}
%Y A210030 Cf. A080015, A080054, A208850.
%K A210030 sign
%O A210030 0,2
%A A210030 _Michael Somos_, Mar 16 2012