Let's Play Osu!

### 题目描述 你在玩一个叫做 Osu 的游戏（某音游）！我们来简化一下游戏规则。一局游戏中需要点击 $n$ 次。每一次点击有两种结果：正确或失误。我们定义正确为符号 `O`，失误为 `X`，那么整局游戏可以被写成一个长度为 $n$ 的由字符 `X` 或 `O` 组成的字符串。 用这个字符串你可以用以下的方法计算游戏的得分：对于一个极大的连续的 `O` 连击，将连击的次数的平方加入到总得分中（即连续的 `O` 的个数的平方）。举例说明，如果你的游戏序列为 `OOXOOOXXOO`，那么极大连续的 `O` 连击共有三个：`OO`，`OOO`，`OO`，所以你的总得分为 $2^2 + 3^2 + 2^2= 17$ 。如果整局游戏里没有一次成功的点击那么总得分就为 $0$。 你现在知道了第 $i$ 次 ($1\leq i\leq n$)点击成功的概率为 $p_i$，换句话说，字符串中第 $i$ 个字符有 $p_i$ 的概率成为 `O`，有 $1-p_i$ 的概率成为 `X`，你的任务是算出你游戏总得分的期望值。 ### 输入格式 第一行包含一个整数表示点击的次数 $n$ $( 1\leq n\leq 10^5 )$ 第二行包含 $n$ 个由空格间隔开的实数 $p_1,p_2,...,p_n ( 0\leq p_i\leq 1 )$ 输入的 $p_i$ 最多为六位小数。 ### 输出格式 输出一个实数——你游戏的期望得分。你的答案必须与标准答案的误差不超过 $10^{-6}$。 ### 样例一说明 $3$ 位字符串一共有 $8$ 种不同的情况。每一种出现的概率为 $0.125$ 所以期望得分是 $\dfrac{(9+4+2+1+4+1+1)}{8}=2.75$。 感谢@[凄魉](/user/45876) 提供的翻译。

You're playing a game called Osu! Here's a simplified version of it. There are $ n $ clicks in a game. For each click there are two outcomes: correct or bad. Let us denote correct as "O", bad as "X", then the whole play can be encoded as a sequence of $ n $ characters "O" and "X". Using the play sequence you can calculate the score for the play as follows: for every maximal consecutive "O"s block, add the square of its length (the number of characters "O") to the score. For example, if your play can be encoded as "OOXOOOXXOO", then there's three maximal consecutive "O"s block "OO", "OOO", "OO", so your score will be $ 2^{2}+3^{2}+2^{2}=17 $ . If there are no correct clicks in a play then the score for the play equals to $ 0 $ . You know that the probability to click the $ i $ -th $ (1<=i<=n) $ click correctly is $ p_{i} $ . In other words, the $ i $ -th character in the play sequence has $ p_{i} $ probability to be "O", $ 1-p_{i} $ to be "X". You task is to calculate the expected score for your play.

The first line contains an integer $ n $ ( $ 1<=n<=10^{5} $ ) — the number of clicks. The second line contains $ n $ space-separated real numbers $ p_{1},p_{2},...,p_{n} $ $ (0<=p_{i}<=1) $ . There will be at most six digits after the decimal point in the given $ p_{i} $ .

Print a single real number — the expected score for your play. Your answer will be considered correct if its absolute or relative error does not exceed $ 10^{-6} $ .

3
0.5 0.5 0.5

2.750000000000000

4
0.7 0.2 0.1 0.9

2.489200000000000

5
1 1 1 1 1

25.000000000000000

For the first example. There are 8 possible outcomes. Each has a probability of 0.125. - "OOO" $ → $ $ 3^{2}=9 $ ; - "OOX" $ → $ $ 2^{2}=4 $ ; - "OXO" $ → $ $ 1^{2}+1^{2}=2 $ ; - "OXX" $ → $ $ 1^{2}=1 $ ; - "XOO" $ → $ $ 2^{2}=4 $ ; - "XOX" $ → $ $ 1^{2}=1 $ ; - "XXO" $ → $ $ 1^{2}=1 $ ; - "XXX" $ → $ $ 0 $ . So the expected score is