<s>[INST] Proving two binomial identities I would like to show that
\begin{align}
&\sum_{j=n-k}^n\binom nj(1-x)^{n-j-1}x^{j-1}(j-nx)\\
&\qquad=\binom n{n-k}(n-k)(1-x)^kx^{n-k-1}\sum_{k=0}^{n-1}\frac{(-1)^k}n\binom{n-1}k\binom n{n-k}(n-k)(1-x)^kx^{n-k-1}\\
&\qquad=(-1)^{n-1}\sum_{k=0}^{n-1}\binom{n-1}k\binom{n+k-1}k(-x)^k
\end{align}
I feel I exhausted all identities/properties of binomials without success. Mathematica says it is true, but how to show it?
 [/INST] The first Identity follows by induction. When $k=0$
\begin{eqnarray*}
\binom{n}{n} \frac{x^{n-1}}{1-x} (n-nx) = \binom{n}{n} nx^{n-1}.
\end{eqnarray*}
Now assume the sum for $k$ and add the $(k+1)^{th}$ term ($j=n-k-1$)
\begin{eqnarray*}
\binom{n}{n-k}(n-k)(1-x)^kx^{n-k-1}+\binom{n}{n-k-1}(1-x)^{k}x^{n-k-2}(n-k-1-nx) = 
\binom{n}{n-k-1}x^{n-k-2}(1-x)^k \left( (k+1)x +(n-k-1)-nx \right) =\binom{n}{n-k-1}(n-k-1)(1-x)^{k+1}x^{n-k-2}.
\end{eqnarray*}
</s>