[seqfan] Limit A209298(n)/n!

[Paul D Hanna]

Seqfans, 
    Sequence http://oeis.org/A209298 is defined by 
 
E.g.f.: Product_{n>=1} (cos(x^n/n) + sin(x^n/n)). 
 
which is a variant of the trivial identity: 
  Product_{n>=1} (cosh(x^n/n) + sinh(x^n/n)) = 1/(1-x). 
 
I ask, What is the limit A209298(n)/n! = ? 
 
Example: 
A209298(1000)/1000! = 0.2942615679517020268...
A209298(2000)/2000! = 0.2939735835938621667...
A209298(3000)/3000! = 0.2938768494981674721...
A209298(4000)/4000! = 0.2938283311328618257...
A209298(5000)/5000! = 0.2937991678075013564...
A209298(6000)/6000! = 0.2937797033327244435...
 
Can anyone evaluate this limit more accurately?  
  
Thanks, 
    Paul