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Problem 638

Weighted lattice paths

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.

Denote $A(P_{a,b})$ to be the area under the path. For the example of a $P_{4,3}$ path given below, the area equals $6$.

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

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

Calculate $\sum_{k=1}^7 C(10^k+k,10^k+k,k)$. Give your answer modulo $1\ 000\ 000\ 007$.

带权格阵路径

• 路径从$(0,0)$开始，到$(a,b)$结束。
• 路径只能沿着格阵每次向上或向右走一步；也就是说，每一步都只能使横、纵坐标增加而不能减少。

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