I was reading about Eulerian polynomials on OEIS, and there is a recurrence given for them, namely: $ C_0(t)=1 $ and C_n(t)=t(1-t)C'_{n-1}(t)+ntC_{n-1}(t)\qquad (n\geq 1).
How can this recurrence relation be derived? I'm taking the definition of Euler polynomials to be $C_n(t)=\sum_{\pi\in S_n}t^{1+d(\pi)}$, where $d(\pi)$ is the number of descents of $\pi$, as can be found on page 22 of Stanley's Enumerative combinatorics. I read of them there during my self-study, but found this recurrence indepedently on OEIS, so I assume there are equivalent.
Thanks.