P1654 OSU!

Original "Product Sorting", see P2577.

osu is a popular casual game. We can simplify and adapt its rules as follows: There are $n$ operations. Each operation is either a success or a failure, where success is denoted by $1$ and failure by $0$. The $n$ operations form a binary string of length $n$. In this string, every maximal run of $X$ consecutive $1$'s contributes $X^3$ points to the score (i.e., these $X$ ones must not be contained within a longer run of consecutive $1$'s; see the sample explanation). Given $n$ and the success probability of each operation, output the expected score, rounded to 1 decimal place.

The first line contains a positive integer $n$, the number of operations. Each of the next $n$ lines contains a real number in $[0, 1]$, denoting the success probability of each operation.

Output a single real number, the answer, rounded to 1 decimal place.

[Sample explanation] $000$ has a score of $0$，$001$ has a score of $1$，$010$ has a score of $1$，$100$ has a score of $1$，$101$ has a score of $2$，$110$ has a score of $8$，$011$ has a score of $8$，$111$ has a score of $27$，the total is $48$, and the expectation is $\dfrac{48}8 = 6.0$. Constraints: $n \leq 1 \times 10^5$. Translated by ChatGPT 5