<s>[INST] Why does this relation of binomials hold? Why does $\sum_{m=n}^N m\binom {m-1}{n-1}=\sum_{m=n}^N n \binom mn=n \binom{N+1}{n+1}$ is true? Is there some special formula for it?
 [/INST] The first equality is due to the equality in @OlivierOloa's answer.  The second one is a bit trick to obtain.


*

*Change $\binom nn$ to $\binom {n+1}{n+1}$.

*Group the first two term.

*Apply the equality $\binom nk + \binom n{k+1} = \binom {n+1}{k+1}$ to condense the leftmost two terms into one.

*Repeat (3) until only one term left.


\begin{align}
&\binom nn + \binom {n+1}n + \binom {n+2}n + \dots + \binom Nn \\
&= \left[\binom {n+1}{n+1} + \binom {n+1}n \right] + \binom {n+2}n + \dots + \binom Nn \\
&= \left[\binom {n+2}{n+1} + \binom {n+2}n\right] + \dots + \binom Nn \\
&= \cdots \\
&= \binom{N+1}{n+1}
\end{align}
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