This problem itself, admittedly, is not a research problem; but rather an intermediate step I've encountered in my research.

Let $(X_i:1\le i\le N)$ be a multivariate normal random vector where i) each coordinate $X_i$ is standard normal and ii) $\mathbb{E}[X_iX_j]=\rho$ for every $1\le i<j\le N$.

My question. Is there a symbolic expression (as a function of $N$ and $\rho$ only) for the following probability: $$ \mathbb{P}\left(X_1\ge 0,X_2\ge 0,\dots,X_N\ge 0\right). $$ Some further notes:

- For $N=2$, using the fact $X_1$ and $\frac{X_2-\rho X_1}{\sqrt{1-\rho^2}}$ are i.i.d. standard normal, one can indeed reach to $\frac14+\frac{\sin^{-1}\rho}{2\pi}$.
- For $N=3$, such formulas are available in standard textbooks, and is given by $\frac18+\frac{3\sin^{-1}\rho}{4\pi}$.

Situation gets more involved beyond $N\ge 4$.

**Edit.** Looking at the structure of the solution for $N=2$ and $N=3$, I postulate that the answer is
$$
\frac{1}{2^N}+\binom{N}{2}\frac{\sin^{-1}\rho}{\eta}
$$
where
$$
\eta = \binom{N}{2}\pi \frac{2^{N-1}}{2^{N-1}-1}.
$$
My rationale is a) for $\rho=0$ it must be $2^{-N}$ as everybody is independent, b) the term, $\sin^{-1}\rho$, should appear exactly once for each pair $1\le i<j\le N$, and c) for $\rho=1$, it should be $\frac12$.

This obviously simplifies to $$ 2^{-N}+\frac{\sin^{-1}\rho\left(2^{N-1}-1\right)}{2^{N-1}\pi} $$