Increasing Sum of Binomial Coefficients

Theorem
Let $n \in \Z$ be an integer.

Then:
 * $\displaystyle \sum_{j \mathop = 0}^n j \binom n j = n 2^{n-1}$

where $\displaystyle \binom n k$ denotes a binomial coefficient.

That is:
 * $\displaystyle 1 \binom n 1 + 2 \binom n 2 + 3 \binom n 3 + \cdots + n \binom n n = n 2^{n-1}$