$$|a_n|\approx\frac{\Gamma(n/2+1)}{\Gamma(n+1)}\exp\left(-\frac{\pi\sqrt{2n}}{2}\right).$$ Ref. Szego, Orthogonal polynomials, AMS. 1959, formula (8.23.4), see also (9.2.9).
Alexandre Eremenko
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