Determining the pattern in a basic numerical series

Question

In the context of finding the series solution for a variable coefficient ODE, I have to determine if a consistent pattern exists between the following numbers:

$\frac{-1} {3!},\frac{-1} {5!},\frac{-3} {7!}, \frac{-15} {9!}\frac{-105} {11!}$

I really cant spot it... any hints?

EDIT

it looks like Im looking for a function of the type "odd number factorial" Since:

$-1*1=-1$

$-1*3=-3$

$-3*5=-15$

$-3*5*7=-105$

However I have no idea how to generate this....

Answer 1

As you noticed, the numerators correspond to double factorial of odd numbers and the denominators to factorial of odd numbers.

So the general term of your numbers seems to be $$-\frac{(2 n-3)\text{!!}}{(2 n+1)!}$$