An accurate asymptotic estimate for the Taylor coefficients was obtained via the saddle point method in
Griffin+Ono+Rolen+Zagier, Jensen polynomials for the Riemann zeta function and other sequences - arXiv 1902.07321 www.pnas.org/cgi/doi/10.1073/pnas.1902572116
A short verification of the asymptotic formula was posted to the Pari/GP users mailing list, with references
Learning with GP: Griffin,Ono,Rolen,Zagier asymptotic formula for xi(s) http://pari.math.u-bordeaux.fr/archives/pari-users-1908/msg00022.html
J. Gélinas