Problem 746

A Messy Dinner

$n$ families, each with four members, a father, a mother, a son and a daughter, were invited to a restaurant. They were all seated at a large circular table with $4n$ seats such that men and women alternate.

Let $M(n)$ be the number of ways the families can be seated such that none of the families were seated together. A family is considered to be seated together only when all the members of a family sit next to each other.

For example, $M(1)=0$, $M(2)=896$, $M(3)=890880$ and $M(10) \equiv 170717180 \pmod {1\ 000\ 000\ 007}$.

Let $S(n)=\displaystyle \sum_{k=2}^nM(k)$.

For example, $S(10) \equiv 399291975 \pmod {1\ 000\ 000\ 007}$.

Find $S(2021)$. Give your answer modulo $1\ 000\ 000\ 007$.

麻烦的晚餐

$n$个四口之家受邀到一家餐厅就餐，每个家庭都由父亲、母亲、儿子、女儿组成。他们围坐在一张有$4n$个座位的圆桌边，且男性和女性总是交替就座。

我们进一步要求同一家庭的四名成员不能坐在一起，并记所有满足条件的就座方式数为$M(n)$。

例如，$M(1)=0$，$M(2)=896$，$M(3)=890880$，$M(10) \equiv 170717180 \pmod {1\ 000\ 000\ 007}$。

记$S(n)=\displaystyle \sum_{k=2}^nM(k)$。

例如，$S(10) \equiv 399291975 \pmod {1\ 000\ 000\ 007}$。

求$S(2021)$，并将你的答案对$1\ 000\ 000\ 007$取余。