  The Motzkin numbers $M_n=\sum_{k=0}^n\binom n{2k}\binom{2k}k/(k+1)$
$(n=0,1,2,\ldots)$ and the central trinomial coefficients $T_n$
($n=0,1,2,\ldots)$ given by the constant term of $(1+x+x^{-1})^n$, have many
combinatorial interpretations. In this paper we establish the following
surprising arithmetic properties of them with $n$ any positive integer:
$$\frac2n\sum_{k=1}^n(2k+1)M_k^2\in\mathbb Z,$$
$$\frac{n^2(n^2-1)}6\,\bigg|\,\sum_{k=0}^{n-1}k(k+1)(8k+9)T_kT_{k+1},$$ and
also $$\sum_{k=0}^{n-1}(k+1)(k+2)(2k+3)M_k^23^{n-1-k}=n(n+1)(n+2)M_nM_{n-1}.$$
