Explicit definition of a recursive sequence

Question

I need to find an explicit definition for the recursive sequence $$a_0=2,\space a_n=a_{n-1}*(2+4n)$$ My first instinct was to do something like $a_n=2(2+4n)^n$. However, that doesn't work because the common ratio isn't $2+4n$ where n is treated as a constant, it changes for each term, meaning the sequence looks like this: $$2*(2+4)*(2+8)*(2+12)*(2+16)*\space...$$ Any ideas?

Answer 1

$$ \prod_{n=1}^j\frac{a_{n}}{a_{n-1}}=\prod_{n=1}^j(2+4n)=2^j\prod_{n=1}^j(2n+1)=2^j(2j+1)!!\\ \implies\frac{a_j}{a_0}=2^j(2j+1)!!\\ \implies a_j=2\frac{(2j+1)!}{j!} $$