## Weighted lattice paths

### Problem 638

Let $P_{a,b}$ denote a path in a $a\times b$ lattice grid with following properties:

- The path begins at $(0,0)$ and ends at $(a,b)$.
- The path consists only of unit moves upwards or to the right; that is the coordinates are increasing with every move.

Define $G(P_{a,b},k)=k^{A(P_{a,b})}$. Let $C(a,b,k)$ equal the sum of $G(P_{a,b},k)$ over all valid paths in a $a\times b$ lattice grid.

You are given that

- $C(2,2,1)=6$
- $C(2,2,2)=35$
- $C(10,10,1)=184\,756$
- $C(15,10,3) \equiv 880\,419\,838 \mod 1\,000\,000\,007$
- $C(10\,000,10\,000,4) \equiv 395\,913\,804 \mod 1\,000\,000\,007$