Summing n times binomial(n,k) I'm trying to do $\sum_{n=a}^b \left( \begin{array}{rl} n \\ a \end{array} \right) n $ . 
Is there a formula, that anybody knows?

We have, just by counting how many times each binomial term appears, that
$$
\sum_{n=a}^b  n \binom{n}{a}= b\sum_{n = a}^b\binom{n}{a} -\left( \sum_{n = a}^{b-1}\binom{n}{a}+\sum_{n = a}^{b-2} \binom{n}{a} + \cdots + \sum_{n = a}^a\binom{n}{a}\right)
$$
For instance, the term $\binom{b}{a}$ appears $b$ times in the original sum, and it appears $b$ times on the right-hand side as well. The term $\binom{a}{a}$ appears $a$ times in the original sum, and on the right-hand side it appears $b$ times, and is then subtracted $b-a$ times, which makes it a total of $a$ times.
Then, using the identity $\sum_{i = k}^l\binom{i}{k} = \binom{l+1}{k+1}$ multiple times, this gives us
$$
b\sum_{n = a}^b\binom{n}{a} -\left( \sum_{n = a}^{b-1}\binom{n}{a}+\sum_{n = a}^{b-2} \binom{n}{a} + \cdots + \sum_{n = a}^a\binom{n}{a}\right)\\
= b\binom{b+1}{a+1} - \left(\binom{b}{a+1} + \binom{b-1}{a+1}+\cdots + \binom{a+1}{a+1}\right)\\
= b\binom{b+1}{a+1} - \sum_{i = a+1}^b\binom{i}{a+1}\\
= b\binom{b+1}{a+1} - \binom{b+1}{a+2}
$$
