Summations again... but not really

From $(0, 0)$ on the coordinate plane, I want to go to $(m, n)$ by travelling only between lattice points: from $(x, y)$ I can only go to $(x+1, y)$ or $(x, y+1)$ . Let there are $f(m, n)$ ways to do that, in $m+n$ moves.

But, sadly, the question not about just finding $f(m, n)$ .

Let $g(C)=\sum_{j=0}^{C} \frac{1}{f(j, C-j)}$ Let $A=\lim_{C\to\infty}g(C)$

Find the integer part of $1000\sqrt{A}$ .