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A210067 Expansion of (phi(-q) / phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.

%I A210067 #25 Feb 16 2025 08:33:17
%S A210067 1,-4,0,16,0,-56,0,160,0,-404,0,944,0,-2072,0,4320,0,-8648,0,16720,0,
%T A210067 -31360,0,57312,0,-102364,0,179104,0,-307672,0,519808,0,-864960,0,
%U A210067 1419456,0,-2299832,0,3682400,0,-5831784,0,9141808,0,-14194200,0,21842368,0
%N A210067 Expansion of (phi(-q) / phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.
%C A210067 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A210067 G. C. Greubel, <a href="/A210067/b210067.txt">Table of n, a(n) for n = 0..1000</a>
%H A210067 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A210067 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A210067 Expansion of (eta(q)^2 * eta(q^2) * eta(q^8)^2 / eta(q^4)^5)^2 in powers of q.
%F A210067 Euler transform of period 8 sequence [ -4, -6, -4, 4, -4, -6, -4, 0, ...].
%F A210067 a(2*n) = 0 unless n=0. a(2*n + 1) = -4 * A001938(n) = -A127393(n).
%F A210067 a(n) = (-1)^n * A134746(n).
%F A210067 Convolution inverse of A131126. Convolution square of A210030.
%F A210067 Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = -32 - 24*sqrt(2) + 4*sqrt(140+99*sqrt(2)). - _Simon Plouffe_, Mar 02 2021
%e A210067 1 - 4*q + 16*q^3 - 56*q^5 + 160*q^7 - 404*q^9 + 944*q^11 - 2072*q^13 + ...
%t A210067 a[n_] := SeriesCoefficient[(EllipticTheta[3, 0, -q]/EllipticTheta[3, 0, q^2])^2, {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Nov 29 2017 *)
%o A210067 (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)^2, n))}
%Y A210067 Cf. A001938, A127393, A131126, A134746, A210030.
%K A210067 sign
%O A210067 0,2
%A A210067 _Michael Somos_, Mar 16 2012