Summation with factorial powers and binonial numbers

Question

I would like to find a closed form expression for the sum \begin{equation*} S(n,N)\triangleq \sum_{c=0}^n N^{[c]}\,\binom{n}{c} \qquad n\leq N \end{equation*} where, clearly, $N,n$ are integers and $N^{[c]}$ is the (falling) factorial power, i.e. \begin{equation*} N^{[c]}\triangleq \frac{N!}{(N-c)!} \end{equation*} I was thinking to study the problem by induction on $n$ (or maybe on $N$), but I don't know if it is the right way to obtain the solution. Since I'm a little bit confused, any suggestion is welcome.

Answer 1

If I do not misread it, making the problem more general $$S(n,N,x)= \sum_{c=0}^n \frac{N!}{(N-c)!}\,\binom{n}{c}\,x^c$$ $$S(n,N,x)=(-1)^n\,x^n\,U\left(-n,N-n+1,-\frac{1}{x}\right)$$ where appears Tricomi's confluent hypergeometric function.