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A210067 Expansion of (phi(-q) / phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function. 3

%I

%S 1,-4,0,16,0,-56,0,160,0,-404,0,944,0,-2072,0,4320,0,-8648,0,16720,0,

%T -31360,0,57312,0,-102364,0,179104,0,-307672,0,519808,0,-864960,0,

%U 1419456,0,-2299832,0,3682400,0,-5831784,0,9141808,0,-14194200,0,21842368,0

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

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A210067/b210067.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of (eta(q)^2 * eta(q^2) * eta(q^8)^2 / eta(q^4)^5)^2 in powers of q.

%F Euler transform of period 8 sequence [ -4, -6, -4, 4, -4, -6, -4, 0, ...].

%F a(2*n) = 0 unless n=0. a(2*n + 1) = -4 * A001938(n) = -A127393(n).

%F a(n) = (-1)^n * A134746(n).

%F Convolution inverse of A131126. Convolution square of A210030.

%F 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 1 - 4*q + 16*q^3 - 56*q^5 + 160*q^7 - 404*q^9 + 944*q^11 - 2072*q^13 + ...

%t 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 (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 Cf. A001938, A127393, A131126, A134746, A210030.

%K sign

%O 0,2

%A _Michael Somos_, Mar 16 2012

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Last modified August 5 14:14 EDT 2021. Contains 346469 sequences. (Running on oeis4.)