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, $S(4,4,15) \bmod 10^9+7 = 325951319$.
Find $S(9,10,1112131415) \bmod 10^9+7$.
Solution
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