Asymptotics of the number of threshold functions and the singularity probability of random $\{\pm1\}$-matrices

Two results concerning the number $P(2,n)$ of threshold functions and the singularity probability $\mathbb{P}_n$ of random $(n\times n)$ $\{\pm1\}$-matrices are established. The following asymptotics are obtained:
$$ P(2,n)\sim2\binom{2^n-1}{n}\text{ and }\mathbb{P}_n\sim n^2\cdot2^{1-n}\quad n\to\infty. $$