Formula with binomial coefficients

Question

There exist any kind formula for the next sum?

$$ \sum_{i = 0}^n \binom{n}{i}i^\alpha $$ with $\alpha$ an integer.

Answer 1

Note that:

$\begin{align*} \sum_{0 \le i \le n} \binom{n}{i} z^i &= (1 + z)^n \\ z \frac{\mathrm{d}}{\mathrm{d} z} \sum_{0 \le i \le n} \binom{n}{i} z^i &= \sum_{0 \le i \le n} \binom{n}{i} i z^i \\ &= z \frac{\mathrm{d}}{\mathrm{d} z} (1 + z)^n \\ &= z n (1 + z)^{n - 1} \end{align*}$

If you evaluate the last expression at $z = 1$ you get:

$\begin{align*} 1 \cdot n \cdot 2^{n - 1} &= \sum_{0 \le i \le n} \binom{n}{i} i \end{align*}$

Repeat the above $\alpha$ times, and you are set. I don't think there is a simple, general formula.