Weighted lattice paths

 Published on Sunday, 7th October 2018, 04:00 am; Solved by 330;
Difficulty rating: 40%

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.
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.
crossed lines

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

  • $C(2,2,1)=6$
  • $C(2,2,2)=35$
  • $C(10,10,1)=184\,756$
  • $C(15,10,3) \equiv 880\,419\,838 \mod 1\,000\,000\,007$
  • $C(10\,000,10\,000,4) \equiv 395\,913\,804 \mod 1\,000\,000\,007$
Calculate $\displaystyle\sum_{k=1}^7 C(10^k+k, 10^k+k,k)$ . Give your answer modulo $1\,000\,000\,007$