  We consider orthogonal polynomials $\{p_{n,N}(x)\}_{n=0}^{\infty}$ on the
real line with respect to a weight $w(x)=e^{-NV(x)}$ and in particular the
asymptotic behaviour of the coefficients $a_{n,N}$ and $b_{n,N}$ in the three
term recurrence $x \pi_{n,N}(x) = \pi_{n+1,N}(x) + b_{n,N} \pi_{n,N}(x) +
a_{n,N} \pi_{n-1,N}(x)$. For one-cut regular $V$ we show, using the Deift-Zhou
method of steepest descent for Riemann-Hilbert problems, that the diagonal
recurrence coefficients $a_{n,n}$ and $b_{n,n}$ have asymptotic expansions as
$n \to \infty$ in powers of $1/n^2$ and powers of $1/n$, respectively.
