A139820 - OEIS

%I #32 Mar 12 2021 22:24:45

%S 1,-8,32,-96,256,-624,1408,-3008,6144,-12072,22976,-42528,76800,

%T -135728,235264,-400704,671744,-1109904,1809568,-2914272,4640256,

%U -7310592,11404416,-17626944,27009024,-41047992,61905088,-92681664,137803776,-203554224

%N Expansion of (phi(-q) / 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).

%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).

%H G. C. Greubel, <a href="/A139820/b139820.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^4) / eta(q^2)^3)^4 in powers of q.

%F Expansion of (phi(-q) / phi(q))^2 = (phi(-q^2) / phi(q))^4 = (phi(-q) / phi(-q^2))^4 = (psi(-q) / psi(q))^4 = (chi(-q^2) / chi(q)^2)^4 = (chi(-q) / chi(q))^4 = (chi(-q)^2 / chi(-q^2))^4 = (f(-q) / f(q))^4 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.

%F Expansion of Jacobian elliptic function k'(q) in powers of nome q.

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

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 4 * u - v^2 * (1 + u)^2.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is g.f. for A001938.

%F G.f.: ((Sum_{k} (-x)^k^2) / (Sum_{k} x^k^2))^2 = (Product_{k>0} (1 + x^(2*k)) / (1 + x^k)^2)^4.

%F a(n) = (-1)^n * A014969(n). Convolution inverse of A014969.

%F Empirical : sum(exp(-Pi)^(n-1)*(-1)^(n+1)*a(n),n=1..infinity) = sqrt(2). Simon Plouffe, Feb. 20, 2011.

%F a(n) ~ (-1)^n * exp(Pi*sqrt(2*n)) / (8 * 2^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2017

%F G.f.: exp(-8*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - _Ilya Gutkovskiy_, Apr 19 2019

%e G.f. = 1 - 8*q + 32*q^2 - 96*q^3 + 256*q^4 - 624*q^5 + 1408*q^6 - 3008*q^7 + ...

%t a[ n_] := SeriesCoefficient[ Sqrt[1 - InverseEllipticNomeQ [ q]], {q, 0, n}];

%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] / EllipticTheta[ 3, 0, q])^2, {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A) / eta(x^2 + A)^3)^4, n))};

%Y Cf. A001938, A014969.

%K sign

%O 0,2

%A _Michael Somos_, May 01 2008