Determine the form of the sequence {9,144,3600,129600,6350400,...}

Question

I'm having a difficult time figuring out the functional form of this sequence:

$$\{9,144,3600,129600,6350400,...\}$$

I'm trying to determine the recursive relationship for a differential equation using power series solutions.

Any help or recommendations are appreciated.

Thank you!

Answer 1

$$a(n)=\left(\frac{(n+2)!}{2}\right)^2$$ or $$a_{n+1}=a_n(n+3)^2$$

Answer 2

We have $$\Big\{9\quad\Big|\quad9\times 16\quad\Big|\quad9\times 16\times 25\quad\Big|\quad 9\times 16\times 25\times 36\quad\Big|\quad 9\times 16\times 25\times 36\times 49\quad\Big|\quad \text{and so on ...}\Big\}$$ therefore$$a_1=9\\a_n={\Big[(n+2)!\Big]^2\over 4}$$