Problem 968

5D Summation

Define
$$P(X_{a,b},X_{a,c},X_{a,d},X_{a,e},X_{b,c},X_{b,d},X_{b,e},X_{c,d},X_{c,e},X_{d,e})$$
as the sum of $2^a3^b5^c7^d11^e$ over all quintuples of non-negative integers $(a, b, c, d, e)$ such that the sum of each two of the five variables is restricted by a given value. In other words, $a+b \le X_{a,b}$, $a+d \le X_{a,d}$, $b+e \le X_{b,e}$ etc.

For example, $P(2,2,2,2,2,2,2,2,2,2)=7120$ and $P(1, 2, 3, 4, 5, 6, 7, 8, 9, 10) \equiv 799809376 \pmod{10^9 + 7}$.

Define a sequence $A$ as follows:

$A_0 = 1$, $A_1 = 7$;
$A_n =(7A_{n−1}+A_{n-2}^2) \bmod(10^9+7)$ for $n \ge 2$.

Also define $Q(n) = P(A_{10n}, A_{10n+1}, A_{10n+2}, \dots , A_{10n+9})$.

Find $\displaystyle\sum_{0 \le n \lt 100}Q(n)$. Give your answer modulo $10^9+7$.

五维求和

定义
$$P(X_{a,b},X_{a,c},X_{a,d},X_{a,e},X_{b,c},X_{b,d},X_{b,e},X_{c,d},X_{c,e},X_{d,e})$$
为所有满足以下条件的非负整数五元组$(a, b, c, d, e)$对应的$2^a3^b5^c7^d11^e$之和：这五个变量中，任意两个变量的和都不能超过对应函数参数；也就是说，$a+b \le X_{a,b}$，$a+d \le X_{a,d}$，$b+e \le X_{b,e}$，依此类推。

例如，$P(2,2,2,2,2,2,2,2,2,2)=7120$，$P(1, 2, 3, 4, 5, 6, 7, 8, 9, 10) \equiv 799809376 \pmod{10^9 + 7}$。

定义数列$A$如下：

$A_0 = 1$，$A_1 = 7$；
对于$n \ge 2$，$A_n =(7A_{n−1}+A_{n-2}^2) \bmod(10^9+7)$。

同时定义$Q(n) = P(A_{10n}, A_{10n+1}, A_{10n+2}, \dots , A_{10n+9})$。

求$\displaystyle\sum_{0 \le n \lt 100}Q(n)$，并对$10^9+7$取余作为你的答案。