  We study the entire functions $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ with
non-monotonic second quotients of Taylor coefficients, namely, such that
$\frac{a_{2m-1}^2}{a_{2m-2}a_{2m}} = a>1$ and
$\frac{a_{2m}^2}{a_{2m-1}a_{2m+1}} = b>1$ for all $m \in \mathbb{N}.$ We obtain
necessary and sufficient conditions under which such functions belong to the
Laguerre-P\'olya I class.
