Given an integer n, let G(n) be the number of integer sequences n − 1 ≥ d₁ ≥ d₂ ≥ · · · ≥ d_n ≥ 0 that are the degree sequence of some graph. We show that G(n) = (c + o(1))4ⁿ/n³/⁴ for some constant c > 0, improving both the previously best upper and lower bounds by a factor of n¹/⁴ (up to polylog-factors). Additionally, we answer a question of Royle, extend the values of n for which G(n) is known exactly from n ≤ 290 to n ≤ 1651 and determine the asymptotic probability that the integral of a (lazy) simple symmetric random walk bridge remains non-negative.