A133553 - OEIS

%I #8 Jul 16 2014 16:52:57

%S 0,1,0,3,12,120,1290,17409,277592,5083659,105675030,2452220144,

%T 62891640900,1766131052829,53900956145218,1776400037307315,

%U 62874491729108656,2378684861565934468,95790461019732936558

%N E.g.f. satisfies: A(x) = x*(sec(exp(A(x))-1)).

%H Alois P. Heinz, <a href="/A133553/b133553.txt">Table of n, a(n) for n = 0..100</a>

%F 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

%p 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);

%t CoefficientList[InverseSeries[Series[x*Cos[1 - E^x],{x,0,20}],x],x] * Range[0,20]! (* _Vaclav Kotesovec_, Jul 16 2014 *)

%K nonn

%O 0,4

%A _Alois P. Heinz_, Aug 27 2008