Python 3.8+,  45  39 bytes

-2 thanks to xnor (while print(...)!=0: → while[print(...)]:)
-4 thanks to Neil ([a*n,a//n][a>n] → a//n or a*n)

a=n=1
while[print(a:=a//n or a*n)]:n+=1

A full program which prints \$a(n)\$ for all natural numbers.

Try it online!

As a recursive function, 53:

f=lambda v,n=1,a=1:a*(v<n)or f(v,n+1,[a*n,a//n][a>n])