(Analysis by Claire Zhang and Brandon Wang)

Let's annotate each character in the route with a subscript that indicates which $i.5$ point it passes. That is, we denote $i+1\to i$ moves by $L_i$, and $i\to i+1$ moves by $R_i$. Then, our string must contain exactly $B_i = \frac{A_i}{2}$ $L_i$'s, and $B_i$ $R_i$'s. In any route with minimal turns, we must have:

In addition, we note that in any route, the final $L_{i+1}$ must be followed by an $L_i$ (and not an $R_{i+1}$) since otherwise Bessie would not have a way to return to $0$.

We claim that this is the only restriction. That is, to count the number of paths, for each $i = 0, 1, \ldots, N-2$ it suffices to count the number of ways to pick which $R_i$'s that are followed by $R_{i+1}$'s if $B_i \geq B_{i+1}$, or which $L_{i+1}$'s followed by $L_i$'s if $B_i \leq B_{i+1}$ (such that the last $L_{i+1}$ is followed by an $L_i$). Then, any such assignment will produce a unique valid path.

Uniqueness is clear, but to show validity, suppose we construct the route by following the assignments, where the route ends when the last $L_0$ is reached (and all previous $L_0$'s are followed by $R_0$'s). We need to check that all of the $L_i$'s and $R_i$'s are actually used; since the number of $R_i$'s and $L_i$'s is the same, we need to check that this path visits $B_i$ $L_i$'s.

We will do this by induction, where $L_0$ is true by assumption. For the inductive step, suppose $B_i$ $L_i$'s appear in the path. Then, if $B_i \geq B_{i+1}$, then $B_i-B_{i+1}$ $L_i$'s are preceded by $R_i$'s, and $B_{i+1}$ of them are preceded by $L_{i+1}$'s. So, in order for all the $L_i$'s to appear, all the $L_{i+1}$'s must also appear. Conversely, if $B_i \leq B_{i+1}$, then since the last $L_{i+1}$ is immediately followed by an $L_i$, if not all $L_{i+1}$'s appear then not all $L_i$'s can appear, contradicting the inductive hypothesis. So, the constructed path contains $B_i$ $L_i$'s for each $i$, and thus crosses $i.5$ exactly $2B_i = A_i$ times. Minimality (i.e. the fact that exactly $(B_0 - 1) + \left(\sum_{i=0}^{n-2} |B_i - B_{i+1}|\right) + (B_{n-1})$ turns are made) follows by the construction.

Now, if $B_i \geq B_{i+1}$, then the number of assignments is just $\binom{B_i}{B_{i+1}}$. In the second the other case, since the last $L_{i+1}$ must be followed by $L_i$, the answer is $\binom{B_{i+1}-1}{B_i-1}$.

Thus we obtain our answer as a product of binomial coefficients:

$$ \prod_{i=0}^{N-2} \begin{cases} \binom{B_i}{B_{i+1}} & \text{ if } B_i \geq B_{i+1} \\ \binom{B_{i+1}-1}{B_i-1} & \text{ otherwise } \end{cases} $$

Let $T = \max_i A_i$. We can precompute factorials in $O(T)$ time. We can compute inverse factorials by first computing the modular inverse of $T!$ (e.g., by raising it to $MOD-2$ with binary exponentiation). Then we can obtain all smaller inverse factorials in decreasing order. Now we can compute each binomial coefficient in the desired expression in $O(1)$ time, for a total runtime of $O(\log(MOD) + T+N)$.

Python solution:

P = int(1e9+7)
MAX_A = int(1e6+1)
# computes a^n mod P
def exp(a, n):
    if n == 0:
        return 1
    base = exp((a*a)%P, n // 2)
    return base if n%2 == 0 else (a*base)%P
# initialize
facts = [1]
for i in range(1, MAX_A):
    facts.append((facts[-1] * i)%P)
inv_facts = [exp(facts[-1], P-2)]
for i in range(MAX_A-1, 0, -1):
    inv_facts.append((inv_facts[-1] * i)%P)
# binom(n, m) = n!/(m!(n-m)!)
binom = lambda n, m : (inv_facts[m] * (facts[n] * inv_facts[n-m])%P)%P
N = int(input())
A = [int(x) for x in input().split()]
B = [a // 2 for a in A]
ans = 1
for i in range(N-1):
    if B[i] >= B[i+1]:
        ans *= binom(B[i], B[i+1])
        ans *= binom(B[i+1]-1, B[i]-1)
    ans %= P