a(n) ~ n^(n-1) * s / (exp(n) * r^n * sqrt(1+s+(exp(2*s)*s^4)/r^2)), where r = 0.4099354376925387635... and s = 0.5741930515285908458... are roots of the system of equations s*cos(1-exp(s)) = r, 1 + exp(s)*s*tan(1-exp(s)) = 0. - Vaclav Kotesovec, Jul 16 2014
MAPLE
A:= proc(n) option remember; if n=0 then 0 else convert (series (x* (sec (exp(A(n-1))-1)), x=0, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n)*n!: seq (a(n), n=0..24);
