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 $1\ 000\ 000\ 007$.

记$P_{a.b}$为$a\times b$格阵中满足如下性质的路径：

记$A(P_{a,b})$为路径下方区域的面积。以下图给出的一条$P_{4,3}$路径为例，其下方区域的面积为$6$。

记$G(P_{a,b},k)=k^{A(P_{a,b})}$。记 $C(a,b,k)$为$a\times b$格阵中所有满足上述条件的路径所对应的$G(P_{a,b},k)$之和。

已知

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

求$\sum_{k=1}^7 C(10^k+k,10^k+k,k)$，并将答案对$1\ 000\ 000\ 007$取余。