Answer
$1, 3,15,1055,$
$a_{10}= 654,729,075$
Work Step by Step
$n!=n(n-1)(n-2)\cdot...\cdot 2\cdot 1$
$0!=1$
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$a_{1}=\displaystyle \frac{(2\cdot 1)!}{2^{1}\cdot 1!}=1$,
$a_{2}=\displaystyle \frac{(2\cdot 2)!}{2^{2}\cdot 2!}=\frac{5\cdot 4}{4}=3$,
$a_{3}=\displaystyle \frac{(2\cdot 3)!}{2^{3}\cdot 3!}=\frac{6\cdot 5\cdot 4}{8}=15$,
$a_{4}=\displaystyle \frac{(2\cdot 4)!}{2^{4}\cdot 4!}=\frac{8\cdot 7\cdot 6\cdot 5}{16}=105$,
$a_{10}=\displaystyle \frac{(2\cdot 10)!}{2^{10}\cdot 10!}=\frac{20\cdot 19\cdot...\cdot 12\cdot 11}{1024}$
$= 654,729,075$
