I need to find $(a/b) \bmod m$ where
$$m= 500000002$$
(hence $$m = 148721 \times 41 \times 41 \times 2\qquad\text{(prime factorization)}$$ basically I need to find $a_n\binom{2n}{n}$, which satisfies the recursion $$a_n = a_{n-1} \times\left(\frac{4n-2}{n}\right).$$
I am using Chinese remainder theorem to solve but unable to get the correct answer.
Can someone help??