Compute the terms $c_2,c_3,c_4$ of a recursively defined relation $c_k=c_{k-1}+kc_{k-2}+1$ ... Could anyone please tell me if my steps and solution is correct or should I change something? 
Compute the terms $c_2,c_3,c_4$ of a recursively defined relation
\begin{align}
c_k &=c_{k-1}+kc_{k-2}+1 & \text{ for all integers  $k \ge 2$}
\end{align}
where initial conditions are $c_0=1$ and $c_1=2$.

Working:
\begin{align}
k = 2,\\
c_2 &= c_1 + 2c_0 + 1\\
&= 2 + 2(1) + 1\\
&= 5\\
\\
k = 3,\\
c_3 &= c_2 + 3c_1 +1\\
&= 5 + 3(2) +1\\
&= 12\\
\\
k = 4,\\
c_4 &= c_3 + 4c_2+1\\
&= 12 + 4(5) + 1\\
&= 33
\end{align}

Perfect.
You did exactly what you were asked to do with sufficient details.
If you want to spend more time on the problem, you may try to find a formula for $C_n$ in terms of $n$ 
