Binomial sum of a sequence I have the following sequence:
$$
\sum\limits_{k=1}^n k\binom{n-1}{k-1}
$$ 
What is the sum of this sequence.

$\sum_{k=1}^n \binom{n-1}{k-1}\cdot k$
$=\sum_{k=1}^n \binom{n-1}{k-1}+\sum_{k=1}^n \binom{n-1}{k-1}\cdot(k-1)$ 
Now, $ \binom{n-1}{k-1}\cdot(k-1)=\frac{(n-1)!}{\{n-1-(k-1)\}!(k-1)!}\cdot(k-1)$
$=(n-1)\frac{(n-2)!}{\{n-2-(k-2)\}!(k-2)!}=(n-1)\binom{n-2}{k-2}$
$\sum_{k=1}^n \binom{n-1}{k-1}\cdot k$
$=\sum_{k=1}^n \binom{n-1}{k-1}+(n-1)\sum_{k=1}^n\binom{n-2}{k-2}$
$=\sum_{r=0}^{n-1} \binom{n-1}{r}+(n-1)\sum_{s=0}^{n-2}\binom{n-2}{s}$ as $\binom{n}{r}=0$ if $r<0$ or $r>n$
$=(1+1)^{n-1}+(n-1)(1+1)^{n-2}=2^{n-2}(2+n-1)=(n+1)2^{n-2} $
