A210066 - OEIS

%I #33 Mar 12 2021 14:04:25

%S 1,-4,16,-48,128,-312,704,-1504,3072,-6036,11488,-21264,38400,-67864,

%T 117632,-200352,335872,-554952,904784,-1457136,2320128,-3655296,

%U 5702208,-8813472,13504512,-20523996,30952544,-46340832,68901888,-101777112,149403264,-218016640

%N Expansion of (phi(q^2) / phi(q))^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="/A210066/b210066.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) / eta(q^8))^2 * (eta(q^4) / eta(q^2))^7)^2 in powers of q.

%F Euler transform of period 8 sequence [ -4, 10, -4, -4, -4, 10, -4, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A134746.

%F a(n) = (-1)^n * A131126(n). Convolution inverse of A134746. Convolution square of A210065.

%F a(n) ~ (-1)^n * exp(sqrt(2*n)*Pi) / (2^(17/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 17 2017

%F Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 1/2 + sqrt(-4+3*sqrt(2)). - _Simon Plouffe_, Mar 02 2021

%e 1 - 4*q + 16*q^2 - 48*q^3 + 128*q^4 - 312*q^5 + 704*q^6 - 1504*q^7 + 3072*q^8 + ...

%t nmax = 40; CoefficientList[Series[Product[((1 - x^k) / (1 - x^(8*k)))^4 * (1 + x^(2*k))^14, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 17 2017 *)

%t a[n_] := SeriesCoefficient[(EllipticTheta[3, 0, q^2]/ EllipticTheta[3, 0, q])^2, {q, 0, n}]; Table[a[n], {n, 0, 30}] (* _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) / eta(x^8 + A))^2 * (eta(x^4 + A) / eta(x^2 + A))^7)^2, n))}

%Y Cf. A131126, A134746, A210065.

%K sign

%O 0,2

%A _Michael Somos_, Mar 16 2012