Inverse Laplace transform of $\frac{2n!}{2s\cdot(2s+1)\cdot(2s+2)\cdots(2s+n)}$

Question

Find the original f function if it's Laplace transform $F(s)$ is equal to : $$\frac{2n!}{2s\cdot(2s+1)\cdot(2s+2)\cdots(2s+n)}$$

This is looking tricky and I don't know how to start it

The correct answe should be (but it could be wrong it happens in my book)

$$1 - n e^{-\frac{t}{2}} + \frac{1}{2} n(n-1) e^{-t} + \cdots + (-1)^n e^{-\frac{nt}{2}}$$

Answer 1

Hint.

$$ F_0(s) = \frac 1s\\ F_n(s) = \frac{n}{2s+n}F_{n-1}(s),\ \ \ n \gt 0 $$