How many integer solutions ( in terms of integer $a,b,c$) has the equation $$ x \cdot y \cdot z =(a-x) (b-y) (c-z), $$ here $x,y,z>0$ and $a-x,b-y,c-y>0$.
I can find number of solutions for some values $a,b,c$ but I hope there exists a formula or a generating function.
For the solution of the equation.
$$xyz=(a-x)(b-y)(c-z)$$
Solution we write expanding on the multipliers. $c=qt$ And $sn=\frac{q-1}{2}$ ; $pk=\frac{q+1}{2}$
$$x=n(bp-as)$$
$$y=s(ak-bn)$$
$$z=tpk$$