Finding a closed form for a discrete sum using generating functions
The term $t_N = {2 N - i \choose N} x^N$ of the sum is hypergoemtric ($t_{N+1}/t_N$ is a rational function of $N$) which means that the closed form of the sum can be found using Gosper's algorithm, if it exists and is hypergeometric.

When I run the algorithm with Mathematica it says
$$x^i \cdot {}_2F_1(\frac{i+1}{2}, \frac{i+2}{2}; i+1; 4 x)$$
where ${}_2F_1$ is the hypergeometric function. This means that a closed form of the kind you are expecting does not exist.

Supplemental:

Gosper's algorithm computes
$$\sum _{i=0}^n \binom{2 n-i}{n} 2^i = 4^n$$
and then
$$\sum _{n=0}^{\infty } x^n \sum _{i=0}^n \binom{2 n-i}{n} 2^i = \frac{1}{1 - 4 x}.$$