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A216060 Expansion of (phi(q) / phi(q^4))^2 in powers of q where phi() is a Ramanujan theta function.

%I A216060 #21 Feb 16 2025 08:33:18
%S A216060 1,4,4,0,0,-8,-16,0,0,20,56,0,0,-40,-160,0,0,72,404,0,0,-128,-944,0,0,
%T A216060 220,2072,0,0,-360,-4320,0,0,576,8648,0,0,-904,-16720,0,0,1384,31360,
%U A216060 0,0,-2088,-57312,0,0,3108,102364,0,0,-4552,-179104,0,0,6592
%N A216060 Expansion of (phi(q) / phi(q^4))^2 in powers of q where phi() is a Ramanujan theta function.
%C A216060 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A216060 G. C. Greubel, <a href="/A216060/b216060.txt">Table of n, a(n) for n = 0..1000</a>
%H A216060 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A216060 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A216060 Expansion of (eta(q^2)^5 * eta(q^16)^2 / (eta(q)^2 * eta(q^8)^5))^2 in powers of q.
%F A216060 Euler transform of period 16 sequence [ 4, -6, 4, -6, 4, -6, 4, 4, 4, -6, 4, -6, 4, -6, 4, 0, ...].
%F A216060 a(4*n) = 0 unless n=0. a(4*n + 3) = 0. a(4*n + 1) = 4 * A079006(n). a(4*n + 2) = 4 * A001938(n).
%F A216060 Convolution square of A208274.
%F A216060 Empirical: Sum{n>=0} a(n)/exp(Pi*n) = 40 + 28*sqrt(2) - 8*sqrt(48+34*sqrt(2)). - _Simon Plouffe_, Mar 02 2021
%e A216060 1 + 4*q + 4*q^2 - 8*q^5 - 16*q^6 + 20*q^9 + 56*q^10 - 40*q^13 - 160*q^14 + ...
%t A216060 a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]/EllipticTheta[3, 0, q^4])^2, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Dec 04 2017 *)
%o A216060 (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))^2, n))}
%Y A216060 Cf. A001938, A079006, A208274.
%K A216060 sign
%O A216060 0,2
%A A216060 _Michael Somos_, Aug 31 2012