<s>[INST] Computing $2 \binom{n}{0} + 2^2 \frac{\binom{n}{1}}{2} + 2^3 \frac{\binom{n}{2}}{3} + \cdots + 2^{n+1} \frac{\binom{n}{n}}{n+1}$ How can I compute the sum $2 \binom{n}{0} + 2^2 \frac{\binom{n}{1}}{2} + 
2^3 \frac{\binom{n}{2}}{3} + \cdots + 2^{n+1} \frac{\binom{n}{n}}{n+1}$? I think I should expand $(1+ \sqrt{2})^n$ or something like this and then find some kind of linear recurrence, but I'm not sure.
 [/INST] Your expression is $\displaystyle\sum_{k=0}^n \frac{x^{k+1}}{k+1} \binom{n}{k}$ evaluated at $x=2$.
Hint. Can you think of where else $\displaystyle\frac{x^{k+1}}{k+1}$ shows up (specifically, in calculus)?
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