968. 5D Summation

Define

as the sum of $2^a{3}^b{5}^c{7}^d{11}^e$$2^a3^b5^c7^d11^e$ over all quintuples of non-negative integers $(a,b,c,d,e)$$(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}$$a+b \le X_{a,b}$, $a+d\le X_{a,d}$$a+d \le X_{a,d}$, $b+e\le X_{b,e}$$b+e \le X_{b,e}$ etc.

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

Define a sequence $A$$A$ as follows:

• $A_0=1$$A_0 = 1$, $A_1=7$$A_1 = 7$;

• $A_n=(7A_{n-1}+A_{n-2}^2)mod({10}^9+7)$$A_n =(7A_{n−1}+A_{n-2}^2) \bmod(10^9+7)$ for $n\ge 2$$n \ge 2$.

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

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

968. 五维求和

记

为：所有满足

的非负整数五元组 $(a,b,c,d,e)$$(a, b, c, d, e)$ 的 $2^a{3}^b{5}^c{7}^d{11}^e$$2^a3^b5^c7^d11^e$ 之和。

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

按如下规则定义序列 $A$$A$：

• $A_0=1$$A_0 = 1$、$A_1=7$$A_1 = 7$；

• 对诸 $n\ge 2$$n \ge 2$，有 $A_n=(7A_{n-1}+A_{n-2}^2)mod({10}^9+7)$$A_n =(7A_{n−1}+A_{n-2}^2) \bmod(10^9+7)$。

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

求 ${\displaystyle \sum _{0\le n<100}Q(n)}$$\displaystyle\sum_{0 \le n \lt 100}Q(n)$ 模 $({10}^9+7)$$(10^9+7)$ 的值。

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