Let $G_i(N;x)=)=\sum_{n=0}^{N-1} \binom{n}{i}n^i x^n$. From this I got,
$G_i(N;x) =1+x+x\sum_{n=1}^{N-1} \sum_{j=0}^{i+1} \frac{1}{n-i+1} \color{blue}{\binom{n}{i}x^nn^j} \binom{n+1}{j}-\binom{N}{i}N^ix^N \ \cdots (1)$
Here all $i,j,n \in \mathbb{N} \cup \{0 \}$
Now I want to replace the $\color{blue}{ \text{blue color part}}$ by $G_j(N;x)$ to get
$G_i(N;x) =1+x+x{\sum_{n=1}^{N-1}{\frac{1}{n-i+1}}} \sum_{j=0}^{i+1} G_j(N,x) \binom{n+1}{j}-\binom{N}{i}N^ix^N \ \cdots (2)$.
But it is $\sum \binom{n}{i}x^nn^j$ not $\sum \binom{n}{j}x^nn^j$.
Can I overcome the situation by some trick or indexing manipulation ?
Help me