Can we find the closed-form of the series?

Question

I want to calculate the series $$ F(N,g)=\frac{1}{g^N}\sum_{m=0}^{N(g-1)}\Big(\sum_{i=0}^{[m/g]}(-1)^i\binom{N}{i}\binom{N-1+m-gi}{N-1}\Big)^2 $$ where $g=2,3,4,\cdots$, and $N$ is any positive integer. The symbol "[ x ]" denotes the integer part of $x$. Can we find the closed form of it? Any suggestion? I find that it may have the asymptotic behavior $$ F(N,g)\sim \frac{1}{\sqrt{N}}. $$