Minimize $a$, but ensure convergence

Hint: Apply the ratio test.

You'll need to understand the size of $$\frac{(2n+2)(2n+1)}{(n+1)^a}$$ as $n\rightarrow \infty.$

It is well-known and not difficult to prove that $$ \frac{(2n)!}{n!^2}=\binom{2n}{n}\sim\frac{4^n}{\sqrt{\pi n}} \tag{1}$$ hence the absolute convergence of the given series for any $a>2$ also follows from asymptotic comparison. It might be interesting to notice that for $a=3$ we have $$ \sum_{n\geq 0}\binom{2n}{n}\frac{1}{n!} = e^2 I_0(2)\approx\frac{101}{6} \tag{2}$$ with $I_0$ being a modified Bessel function of the first kind, for instance.