Problem Archive

Let $n$ be a positive integer and let $E_n$ be the set of $n$-tuples of strictly positive integers.

For $u = (u_1, \cdots, u_n)$ and $v = (v_1, \cdots, v_n)$ two elements of $E_n$, we define:

• the Sum Of Products of $u$ and $v$, denoted by $\langle u, v\rangle$, as the sum $\displaystyle\sum_{i = 1}^n u_i v_i$;
• the Product Of Sums of $u$ and $v$, denoted by $u \star v$, as the product $\displaystyle\prod_{i = 1}^n (u_i + v_i)$.

Let $R_n(M)$ be the sum of $u \star v$ over all ordered pairs $(u, v)$ in $E_n$ such that $\langle u, v\rangle = M$.
For example: $R_1(10) = 36$, $R_2(100) = 1873044$, $R_2(100!) \equiv 446575636 \bmod 10^9 + 7$.

Find $R_6(10000!)$. Give your answer modulo $10^9+7$.