Is there any way to calculate the central q-Binomial coefficient efficiently.
For example, $$\binom{2n}{n}_1=\frac{(2n)!}{(n!)^2}$$ First few values of $\binom{2n}{n}_2$ are $1,3,35,1395,200787,109221651$
First few values of $\binom{2n}{n}_3$ are $1,4,130,33880,75913222,1506472167928,267598665689058580$
These can be calculated using the function $QBinomial[2n,n,q]$. But this function works for $n\le10^4$. Is there any property of central q-Binomial coefficient that allows fast calculation for larger $n$?
It seems that an asymptotics could be $$\binom{2n}{n}_q \sim C_q\, q^{n^2}$$ Computing the coefficient for $n=1000$, the values are $$\left( \begin{array}{cc} q & C_q \\ 2 & 3.462746619 \\ 3 & 1.785312342 \\ 4 & 1.452353642 \\ 5 & 1.315213556 \\ 6 & 1.241175663 \\ 7 & 1.195035240 \\ 8 & 1.163594397 \\ 9 & 1.140822757 \\ 10 & 1.123582755 \\ 11 & 1.110084028 \\ 12 & 1.099231752 \\ 13 & 1.090319360 \\ 14 & 1.082870737 \\ 15 & 1.076553491 \\ 16 & 1.071128609 \\ 17 & 1.066419860 \\ 18 & 1.062294483 \\ 19 & 1.058650573 \\ 20 & 1.055408622 \end{array} \right)$$
Considering the last term in your lists, this would give $116190496$ and $267965804863721413$.