An accurate asymptotic estimate for the Taylor coefficients was obtained via the saddle point method in

Griffin+Ono+Rolen+Zaghier, Jensen polynomials for the Riemann zeta function and other sequences - arXiv 1902.07321

A short verification of the asymptotic formula was posted to the Pari/GP users mailing list, with references

Learning with GP: Griffin,Ono,Rolen,Zaghier asymptotic formula for xi(s)
http://pari.math.u-bordeaux.fr/archives/pari-users-1908/msg00022.html

J. Gélinas