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Chromatic Conundrum

Problem 544 Published on Saturday, 23rd January 2016, 07:00 pm; Solved by 158;
Difficulty rating: 90%

Let F(r,c,n) be the number of ways to colour a rectangular grid with r rows and c columns using at most n colours such that no two adjacent cells share the same colour. Cells that are diagonal to each other are not considered adjacent.

For example, F(2,2,3) = 18, F(2,2,20) = 130340, and F(3,4,6) = 102923670.

Let S(r,c,n) = $\sum_{k=1}^{n}$ F(r,c,k).

For example, S(4,4,15) mod 109+7 = 325951319.

Find S(9,10,1112131415) mod 109+7.