Recurrence relations of the type $a(n,k) = a(n-1,k-1) a(n,k-1) /a(n-1,k-1)- a(n,k-1) $

Question

How to solve recurrence relations of the type $$a(n,k) = \frac{a(n-1,k-1) a(n,k-1)}{ a(n-1,k-1)}- a(n,k-1) $$ with $a(n,1) = n$?

This was my main equation:

which I have solved to reduce it further but I can't solve it to make a function of it.

Answer 1

This can be easily proven by induction:

(Base case $F(N,2)=N(N-1)$ can be easily checked)

Suppose $F(N,k)=\frac{N!}{(N-k)!(k-1)!}$ by induction hypotheses.

Then $F(N,k+1)=\frac{1}{\frac{(N-k-1)!(k-1)!}{(N-1)!}-\frac{(N-k)!(k-1)!}{N!}}=\frac{1}{(N-(N-k))(\frac{(N-k-1)!(k-1)!}{N!})}=\frac{1}{(\frac{(N-k-1)!k!}{N!})}$
$=\frac{N!}{(N-k-1)!k!}=\frac{N!}{(N-(k+1))!((k+1)-1)!}$.

Hence by induction it works for all natural number $K \geq 2$.