<s>[INST] How can I express the ration of double factorials $\frac{(2n+1)!!}{(2n)!!}$ as a single factorial? How can I change the  double factorial of $$\frac{(2n+1)!!}{(2n)!!}$$ to single factorial?
 [/INST] $$(2n+1)!!=(2n+1)(2n-1)(2n-3) \cdots (3) (1)$$
$$=\frac{(2n+1)(2n)(2n-1)(2n-2)(2n-3)(2n-4) \cdots (4)(3)(2) (1)}{(2n)(2n-2)(2n-4) \cdots (4)(2)}=\frac{(2n+1)(2n)(2n-1)(2n-2)(2n-3)(2n-4) \cdots (4)(3)(2) (1)}{(2)(n)(2)(n-1)(2)(n-2) \cdots (2)(2)(2)(1)}$$
$$=\frac{(2n+1)!}{(2^n)n!}$$
Similarly, 
$$(2n)!!=(2n)(2n-2)(2n-4) \cdots (4)(2)=(2)(n)(2)(n-1)(2)(n-2) \cdots (2)(2)(2)(1)=(2^n)n!$$
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