Summations again... but not really

Discrete Mathematics Level 4

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 \(1000\sqrt{A}\).


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