Let sequences $(c_n)$ and $(d_n)$ be given by $$c_0=0,\:d_0=1$$ and recursively for $n\ge 1$ by $$\begin{align} c_n & =\frac{n}{n+1}c_{n-1}+\frac{2n}{n+1}d_{n-1} \\[2ex] d_n & =2c_{n-1}+d_{n-1} \end{align}$$
I'd like to show that all $c_{n},d_{n}$ are integers. (Creat by wang yong xi）
My try： Since $$\begin{align}(n+1)c_n & =nc_{n-1}+2nd_{n-1}\\[1ex] d_n & =2c_{n-1}+d_{n-1} \end{align}$$ we easily find $$c_{1}=1,\:d_{1}=1,\\ c_{2}=2,\:d_{2}=3,\\ c_{3}=6,\:d_{3}=7,$$ a.s.o. How to prove that all the $c_{n},d_{n}$ are integers?