  A sequence $a=(a_n)_{n=1}^\infty$ of non-negative integers is called
realizable if there is a map $T$ on a set such that $a_n$ is equal to the
number of periodic points of $T$ of (not necessarily exact) period $n$ for all
$n\geqslant1$. In this article, we show that for every
$r_1,r_2,s,t,u\in\mathbb{Z}_{\geqslant0}$ with $r_1+r_2\geqslant1$, the
sequence $(V(n,r_1,r_2,s,t,u))_{n=1}^\infty$ is realizable, where
$$V(n,r_1,r_2,s,t,u)=\sum_{k=0}^n \binom{n}{
k}^{r_1}\binom{n}{2k}^{r_2}\binom{n+k}{k}^s\binom{2k}{k}^t\binom{2(n-k)}{n-k}^u$$
and $0^0=1$. From this, we deduce that many famous combinatorial sequences are
realizable, for example, the sequences of certain Ap\'ery-like numbers, Franel
numbers of any order and central trinomial coefficients; while we also show
that the sequences of the Catalan numbers, Motzkin numbers and Schr\"oder
numbers are even not almost realizable.

Number Theory;Combinatorics;Dynamical Systems