On Motzkin numbers and central trinomial coefficients

The Motzkin numbers $M_n={\sum }_{k=0}^n\left(\genfrac{}{}{0px}{}{n}{2k}\right)\left(\genfrac{}{}{0px}{}{2k}{k}\right)\mathrm{/}(k+1)$ $(n=0,1,2,\dots )$ and the central trinomial coefficients $T_n$ ($n=0,1,2,\dots )$ 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: $$\frac{2}{n}\sum _{k=1}^n(2k+1)M_k^2\in \mathbb{Z},$$ $$\frac{n^2(n^2-1)}{6}\mid \sum _{k=0}^{n-1}k(k+1)(8k+9)T_k{T}_{k+1},$$ and also $$\sum _{k=0}^{n-1}(k+1)(k+2)(2k+3)M_k^2{3}^{n-1-k}=n(n+1)(n+2)M_n{M}_{n-1}\mathrm{.}$$