Figured out, can this be deleted?
The actual sum is this $\displaystyle
\sum\limits_{k = 0}^n {\left( {k + 1} \right)\left( {\begin{array}{*{20}c}
{n + 1} \\
{n - k} \\
\end{array}} \right)} $. You can choose any number form 0 to n letters.
Say n=10; then you may choose 3 a’s and a b along with any six numbers from 1 to 7. In this case, you have chosen 4 letters and 6 numbers. There are 5 ways to pick 4 letters: aaaa, aaab, aabb, abbb, & bbbb. There are 7 ways to pick six numbers from 1 to 7.
Now we sum on all the cases.