Displacement of permutation

Let $p$ be a permutation of the set $\{1,2,3,\dots ,n\}$. We call the sum $D_k(p)=\sum _{i=1}^n|p_i-i{|}^k$ the total displacement with $k$th power of $p$. For example, if $p$ = $(4,2,1,3)$, $D_1(p)=|4-1|+|2-2|+|1-3|+|3-4|=6$, $D_2(p)=|4-1{|}^2+|2-2{|}^2+|1-3{|}^2+|3-4{|}^2=14$, $D_3(p)=|4-1{|}^3+|2-2{|}^3+|1-3{|}^3+|3-4{|}^3=36$.

For any positive integer $n$, assume every possible permutation $p$ of the set $\{1,2,3,\dots ,n\}$ is uniformly distributed. Let $E(n)$ be the expected value of $D_1(p)+D_2(p)+D_3(p)$. You are given $E(2)=3$, $E(3)=40/3$, $E(5)=432/5$, and all $E(n)$ are rational number. Let $F(n)$ be the numerator of $E(n)$ with $E(n)$ be fraction in lowest terms. So, $F(2)=3$, $F(3)=40$, $F(5)=432$ and so on.

Let $S(n)=\sum _{i=1}^{n}F(i)$. Find $S(12345678987654321)\mathrm{mod}1000000007$.

---

Thanks to baihacker for the idea.