Problem #WSP-000401

Problem

In the $n^{\text{th}}$ row of Pascal’s triangle, leave the left $1$ untouched, multiply the next number along (which is $(\genfrac{}{}{0ex}{}{n}{1})=n$) by $2$, multiply the next number along (which is $(\genfrac{}{}{0ex}{}{n}{2})=\frac{n(n-1}{2}$) by $4$, and so on, until you multiply the right-hand $1$ by $2^n$. That is, multiply the $k^{\text{th}}$ number from the left by $2^k$.

Now what’s the sum of the numbers in the $n^{\text{th}}$ row?