{"app":{"locale":"en-GB"},"problem":{"content":"\rLet $P_{a,b}$ 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.

$\"crossed$

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 $1\\,000\\,000\\,007$","id":638,"title":"Weighted lattice paths"},"problems":[{"id":1,"title":"Multiples of 3 and 5"},{"id":2,"title":"Even Fibonacci numbers"},{"id":3,"title":"Largest prime factor"},{"id":4,"title":"Largest palindrome product"},{"id":5,"title":"Smallest multiple"},{"id":6,"title":"Sum square difference"},{"id":7,"title":"10001st prime"},{"id":8,"title":"Largest product in a series"},{"id":9,"title":"Special Pythagorean triplet"},{"id":10,"title":"Summation of primes"},{"id":11,"title":"Largest product in a grid"},{"id":12,"title":"Highly divisible triangular number"},{"id":13,"title":"Large sum"},{"id":14,"title":"Longest Collatz sequence"},{"id":15,"title":"Lattice paths"},{"id":16,"title":"Power digit sum"},{"id":17,"title":"Number letter counts"},{"id":18,"title":"Maximum path sum I"},{"id":19,"title":"Counting Sundays"},{"id":20,"title":"Factorial digit sum"},{"id":21,"title":"Amicable numbers"},{"id":22,"title":"Names 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