Expansion of (eta(q)^2 * eta(q^4) / eta(q^2)^3)^4 in powers of q.
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.
Expansion of Jacobian elliptic function k'(q) in powers of nome q.
Euler transform of period 4 sequence [ -8, 4, -8, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 4 * u - v^2 * (1 + u)^2.
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.
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.
Empirical : Sum_{n>=1} exp(-Pi)^(n-1)*(-1)^(n+1)*a(n) = sqrt(2). -
Simon Plouffe, Feb 20 2011
a(n) ~ (-1)^n * exp(Pi*sqrt(2*n)) / (8 * 2^(1/4) * n^(3/4)). -
Vaclav Kotesovec, Sep 08 2017
G.f.: exp(-8*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). -
Ilya Gutkovskiy, Apr 19 2019
