  Supposing that $A(z)$ is an exponential polynomial of the form
  $$
  A(z)=H_0(z)+H_1(z)e^{\zeta_1z^n}+\cdots +H_m(z)e^{\zeta_mz^n},
  $$ where $H_j$'s are entire and of order $<n$, it is demonstrated that the
function $H_0(z)$ and the geometric location of the leading coefficients
$\zeta_1,\ldots,\zeta_m$ play a key role in the oscillation of solutions of the
differential equation $f''+A(z)f=0$. The key tools consist of value
distribution properties of exponential polynomials, and elementary properties
of the Phragm\'en-Lindel\"of indicator function. In addition to results in the
whole complex plane, results on sectorial oscillation are proved.
