I've found that I can simplify $\prod\limits_{k=1}^{n}2k-1$ to $\frac{(2\cdot n)!}{2^n\cdot n!}$, for example:
$\prod\limits_{k=1}^{4}2k-1=1\cdot3\cdot5\cdot7=\frac{1\cdot2\cdot3\cdot4\cdot5\cdot6\cdot7\cdot8}{2\cdot4\cdot6\cdot8}=\frac{1\cdot2\cdot3\cdot4\cdot5\cdot6\cdot7\cdot8}{(1\cdot2)\cdot(2\cdot2)\cdot(3\cdot2)\cdot(4\cdot2)}=\frac{(2\cdot4)!}{4!\cdot2^4}$
I'm having difficulties applying the same tricks on $\prod\limits_{k=1}^{n}3k+1$.
Any ideas anyone?