[seqfan] E.g.f.s on the Verge of Negativity

[Paul D Hanna]

SeqFans, 
     Please consider the sequences 
http://oeis.org/A232690 : A(x) = exp( 1/A(x) * Integral A(x)^3 dx ),http://oeis.org/A232691 : A(x) = exp( 1/A(x)^2 * Integral A(x)^6 dx ),http://oeis.org/A232692 : A(x) = exp( 1/A(x)^3 * Integral A(x)^8 dx ). 
 
These illustrate the followimg conjecture.   
 
Given  G(x,n,k) = G such that G = exp( 1/G^n * Integral G^k dx ) 
then  G(x,n,k) consists solely of positive coefficients when k >= A047399(n) 
whereA047399 lists numbers that are congruent to {0,3,6} mod 8.    
Could someone test the above conjecture, that A047399 forms positive e.g.f.s,but if k < A047399(n) then G(x,n,k) has at least one negative coefficient?   
   
   Below I give some examples of these positive e.g.f.s that are on the verge of being non-positive. 
Thanks, 
      Paul EXAMPLES of positive e.g.f.s:G(x,1,3): [1, 1, 2, 7, 33, 202, 1495, 13107, 132062, ...] = A232690
G(x,2,6): [1, 1, 3, 19, 161, 1857, 25843, 433891, ...]    = A232691
G(x,3,8): [1, 1, 3, 24, 213, 3096, 46071, 967608, ...]    = A232692
G(x,4,11): [1, 1, 4, 44, 552, 11048, 236576, 6830048, ...] 
G(x,5,14): [1, 1, 5, 70, 1135, 28970, 808345, 29798350, ...] 
G(x,6,16): [1, 1, 5, 81, 1305, 38961, 1077021, 46597185, ...] 
G(x,7,19): [1, 1, 6, 115, 2281, 80094, 2754871, 139456135, ...]
G(x,8,22): [1, 1, 7, 155, 3649, 147673, 6060999, 353192323, ...]
G(x,9,24): [1, 1, 7, 172, 3997, 181816, 7307731, 482076712, ...]
G(x,10,27): [1, 1, 8, 220, 5940, 300460, 14146360, 1032200280, ...]
G(x,11,30): [1, 1, 9, 274, 8423, 469986, 25312045, 2032845166, ...]
G(x,12,32): [1, 1, 9, 297, 9009, 551601, 29056761, 2580921369, ...]
G(x,13,35): [1, 1, 10, 359, 12249, 810986, 48378743, 4623767555, ...]
G(x,14,38): [1, 1, 11, 427, 16177, 1153649, 76741483, 7879744075, ...]
G(x,15,40): [1, 1, 11, 456, 17061, 1313976, 85583151, 9563829336, ...]
G(x,16,43): [1, 1, 12, 532, 21928, 1796232, 129634240, 15338665792, ...]
G(x,17,46): [1, 1, 13, 614, 27631, 2402122, 190017873, 23797591486, ...]
G(x,18,48): [1, 1, 13, 649, 28873, 2680321, 207923221, 28009869193, ...]
G(x,19,51): [1, 1, 14, 739, 35697, 3486478, 295146391, 41660943783, ...]
G(x,20,54): [1, 1, 15, 835, 43505, 4464585, 409135135, 60519094075, ...]
... 
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