How are These Two Infinite Summations Equal?

[donutmax]

$$\sum_{n=0}^{\infty} \frac{2^{n+1}(n+1)t^n}{(n+2)!}=\sum_{n=0}^{\infty} \frac{2^{n}nt^{n-1}}{(n+1)!}$$

How are the above summation equal?

 

[Landau]

Since $\sum_{n=0}^\infty a_n=a_{0}+\sum_{n=0}^\infty a_{n+1}$, and $a_0$ happens to be zero.