Eccoci.

In the context of formal languages, any finite sequence of letters of a given alphabet $\mathrm{\Sigma }$ is called a word over $\mathrm{\Sigma }$. We call a word incomplete if it does not contain every letter of $\mathrm{\Sigma }$.

For example, using the alphabet $\mathrm{\Sigma }=\{a,b,c\}$, '$ab$', '$abab$' and '$$' (the empty word) are incomplete words over $\mathrm{\Sigma }$, while '$abac$' is a complete word over $\mathrm{\Sigma }$.

Given an alphabet $\mathrm{\Sigma }$ of $\alpha$ letters, we define $I(\alpha ,n)$ to be the number of incomplete words over $\mathrm{\Sigma }$ with a length not exceeding $n$.
For example, $I(3,0)=1$, $I(3,2)=13$ and $I(3,4)=79$.

Let ${\displaystyle S(k,n)=\sum _{\alpha =1}^{k}I(\alpha ,n)}$.
For example, $S(4,4)=406$, $S(8,8)=27902680$ and $S(10,100)\equiv 983602076mod1000000007$.

Find $S({10}^7,{10}^{12})$. Give your answer modulo $1000000007$.