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A208274 Expansion of phi(q) / phi(q^4) in powers of q where phi() is a Ramanujan theta function.

%I A208274 #25 Feb 16 2025 08:33:16
%S A208274 1,2,0,0,0,-4,0,0,0,10,0,0,0,-20,0,0,0,36,0,0,0,-64,0,0,0,110,0,0,0,
%T A208274 -180,0,0,0,288,0,0,0,-452,0,0,0,692,0,0,0,-1044,0,0,0,1554,0,0,0,
%U A208274 -2276,0,0,0,3296,0,0,0,-4724,0,0,0,6696,0,0,0,-9408,0,0
%N A208274 Expansion of phi(q) / phi(q^4) in powers of q where phi() is a Ramanujan theta function.
%C A208274 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A208274 Differs from A127391 only at n=0. - _R. J. Mathar_, Mar 18 2012
%H A208274 G. C. Greubel, <a href="/A208274/b208274.txt">Table of n, a(n) for n = 0..1000</a>
%H A208274 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A208274 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A208274 Expansion of eta(q^2)^5 * eta(q^16)^2 / (eta(q)^2 * eta(q^8)^5) in powers of q.
%F A208274 Euler transform of period 16 sequence [ 2, -3, 2, -3, 2, -3, 2, 2, 2, -3, 2, -3, 2, -3, 2, 0, ...].
%F A208274 G.f. A(x) satisfies  A(x)^2 - 2*A(x) + 2 = A134746(x^2), which means (phi(q) / phi(q^4) - 1)^2 + 1 = (phi(q^2) / phi(q^4))^2.
%F A208274 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^2 - 2*u + 2) * (v^2 - 2*v + 2) - v^2.
%F A208274 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = 4 * u * (u - 1) * (2 - u) * v * (v - 1) * (2 - v) - (u - v)^4.
%F A208274 G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = g(t) where q = exp(2 Pi i t) and g() is g.f. for A112128.
%F A208274 a(4*n) = 0 unless n=0. a(4*n + 2) = a(4*n + 3) = 0. a(4*n + 1) = 2 * A079006(n). a(n) = (-1)^n * A208604(n). Convolution inverse is A112128.
%e A208274 1 + 2*q - 4*q^5 + 10*q^9 - 20*q^13 + 36*q^17 - 64*q^21 + 110*q^25 - 180*q^29 + ...
%t A208274 a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]/EllipticTheta[3, 0, q^4], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Dec 04 2017 *)
%o A208274 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^8 + A)^5), n))}
%Y A208274 Cf. A079006, A112128, A134746, A208604.
%K A208274 sign
%O A208274 0,2
%A A208274 _Michael Somos_, Mar 12 2012