My target is to write the expression $(1)$ in terms of $P_k(N,x)$ Let $P_k(N,x)=\sum_{n=0}^{N-1} \binom{2n}{n} n^kx^n$. Suppose we have the expression as follows:
$A=\sum_{n=1}^{N-1} \binom{2n+1}{n+1} (n+1)^kx^{n+1}, \ \cdots (1)$
My target is to write the expression $(1)$ in terms of $P_k(N,x)$ or simply I want to embed $P_k(N,x)$ into $(1)$. 
I have proceed as follows:
\begin{align} A &=\sum_{n=1}^{N-1}\left[ \binom{2n}{n}+\binom{2n}{n+1} \right] (n+1)^kx^{n+1} \\ &=\sum_{n=1}^{N-1} \binom{2n}{n} (n+1)^kx^{n+1}+\sum_{n=1}^{N-1}\binom{2n}{n+1}(n+1)^kx^{n+1} \\ &= \sum_{n=1}^{N-1} \binom{2n}{n} \sum_{l=0}^{k} \binom{k}{l} n^lx^{n+1}+\sum_{n=1}^{N-1}\binom{2n}{n+1}(n+1)^kx^{n+1} \\ &=x\sum_{l=0}^k \binom{k}{l} \sum_{n=1}^{N-1}\binom{2n}{n} n^lx^n+ \sum_{n=1}^{N-1}\binom{2n}{n+1}(n+1)^kx^{n+1} \\ &=x \sum_{l=0}^{k} \binom{k}{l} P_l(N,x)+\color{blue}{\sum_{n=1}^{N-1}\binom{2n}{n+1}(n+1)^kx^{n+1}}\end{align}
So I have been successfully replaced or input $P_k(N,x)$ into the first expression but unable to insert into the 2nd part( blue color part). 
Can someone help me with some trick ?
Can I insert $P_k(N,x)$ in expression $(1)$ without splitting it ?


Let $$P_k(N,x)=\sum_{n=0}^{N-1}\binom{2n}{n}n^kx^n$$      We  obtain
  \begin{align*}
\color{blue}{A}&\color{blue}{=\sum_{n=1}^{N-1}\binom{2n+1}{n+1}(n+1)^kx^{n+1}}\\
&=\sum_{n=2}^{N}\binom{2n-1}{n}n^kx^n\tag{1}\\
&=\sum_{n=2}^{N}\binom{2n-1}{n-1}n^kx^n\tag{2}\\
&=\frac{1}{2}\sum_{n=2}^{N}\binom{2n}{n}n^kx^n\tag{3}\\
&\,\,\color{blue}{=\frac{1}{2}\binom{2N}{N}N^kx^N+\frac{1}{2}P_k(N,x)-x}
\end{align*}

Comment:


*

*In (1) we shift the index by $1$ and  start with $n=2$.

*In (2) we use the  binomial identity $\binom{p}{q}=\binom{p}{p-q}$.

*In (3)  we  use the binomial  identity $\binom{p}{q}=\frac{p}{q}\binom{p-1}{q-1}$.
