[seqfan] A120419 "A mysterious sequence"

[Paul D Hanna]

Seqfans, 
    The "mysterious sequence":  https://oeis.org/draft/A120419 has keyword "obscure" ... 
   
I found that the e.g.f. satisfies the somewhat obtuse formula: 
   
    A(x) = ( 1 + Integral (A(x) * Integral A(x) dx) dx )^2. 
   
The sequence calls for a closed form; can someone derive a closed form for this e.g.f.? 
    
Note that the e.g.f. for Euler numbers (A000364) satisfy: 
 
    G(x) = 1 + Integral (G(x) * Integral G(x)^2 dx) dx  when  G(x) = 1/cos(x)  
 which makes one suspect that there is a closed form for A120419 
(and perhaps a simpler formula than I gave!). 
  
Regards,
   Paul