# 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 $$1000\sqrt{A}$$.

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