Increasing Sum of Binomial Coefficients

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

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

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

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

Proof
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