Find a closed expression for a formula including summation Let:
$$\sum\limits_{k = 0}^n {k\left( {\matrix{
   n  \cr 
   k  \cr 
 } } \right)}  \cdot {4^{k - 1}} \cdot {3^{n - k}}$$
Find a closed formula (without summation). I think I should define this as a "series" which generated by $F(x)$. I don't really have a lead here.
Any ideas? Thanks. 

The binomial expansion is given by
\begin{align}
(1+x)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^{k}.
\end{align}
Differentiating both sides with respect to $x$ yields
\begin{align}
n (1+x)^{n-1} = \sum_{k=0}^{n} k \ \binom{n}{k} x^{k-1}.
\end{align}
Multiplying this last expression by $3^{n} 4^{-1}$ leads to
\begin{align}
\frac{3^{n} n}{4} \ x(1+x)^{n-1} = \sum_{k=0}^{n} k \ \binom{n}{k} 4^{-1} 3^{n} x^{k}.
\end{align}
Now let $x = 4/3$ to obtain the desired expression
\begin{align}
 n \cdot 7^{n-1} = \sum_{k=0}^{n} k \ \binom{n}{k} 4^{k-1} 3^{n-k}.
\end{align}
