Property of the gamma function, and evaluation of a product

Question

Using $\Gamma(n)= (n-1)\Gamma(n-1)$ , I get, $\Gamma(n + \frac{3}{2}) = \frac{\sqrt{\pi}(2n+1)(2n-1)(2n-3)(2n-5)..}{2^{n+1}} $, where there are $N = n+1$ brackets. This can be written as

$$\Gamma(n+\frac{3}{2}) = \frac{\sqrt{\pi}}{2^{n+1}}\prod_{k=1}^{n+1}[2(n-k)+3]$$

Is this reasoning correct, and how do I evaluate this product?

Answer 1

Use $$1\times3\times5\times\cdots\times(2n+1) =\frac{(2n+1)!}{2\times 4\times\cdots\times(2n)} =\frac{(2n+1)!}{2^n n!}$$ etc.