  Let $K$ be an algebraically closed field of characteristic $0$, and let $A$
be an Artinian Gorenstein local commutative and Noetherian $K$--algebra, with
maximal ideal $M$. In the present paper we prove a structure theorem describing
such kind of $K$--algebras satisfying $M^4=0$. We use this result in order to
prove that such a $K$--algebra $A$ has rational Poincar\'e series and it is
always smoothable in any embedding dimension, if $\dim_K M^2/M^3 \le 4$. We
also prove that the generic Artinian Gorenstein local $K$--algebra with socle
degree three has rational Poincar\'e series, in spite of the fact that such
algebras are not necessarily smoothable.
