[seqfan] New? Constant for a Functional Equation

[Paul D Hanna]

Seqfans, 
   It appears that there is a new(?) constant that arises out of the following functional equation. 
Would someone verify and perhaps generate more digits of this constant? 
 
Consider the functional equation: 
 
(*) G(x) = 1 + x * [d/dx x*G(x)^n] / G(x)^k  
 
where n and k are nonnegative integers. 
  
Examples in the OEIS include: 
Ex.1:  n=1, k=0: 
A(x) = 1 + x * [d/dx x*A(x)^1] / A(x)^0 generates A000142 - Factorial numbers:  
1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, ...
Ex.2:  n=1, k=0: 
A(x) = 1 + x * [d/dx x*A(x)^2] / A(x)^1 generates A001147 - Odd double factorials:  
1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425, ...
Ex.3:  n=2, k=2: 
A(x) = 1 + x * [d/dx x*A(x)^2] / A(x)^2 generates A112934:  
1, 1, 2, 6, 26, 158, 1282, 13158, 163354, 2374078, 39456386, ...
Ex.4:  n=3, k=2: 
A(x) = 1 + x * [d/dx x*A(x)^3] / A(x)^2 generates A007559 - Triple factorials:   
1, 1, 4, 28, 280, 3640, 58240, 1106560, 24344320, 608608000, ...
Ex.5: n=3, k=3: 
A(x) = 1 + x * [d/dx x*A(x)^3] / A(x)^2 generates A112936:  
1, 1, 3, 15, 111, 1131, 14943, 243915, 4742391, 106912131, ...
So the functional equation (*) is some worth study. 
  
Note that if k is too large for a fixed n, the A(x) that satisfies (*) will be a power series in x with negative coefficients somewhere in the expansion of A(x). 
   
I ask, what is the largest k for fixed n such that (*) is satisfied by a power series in x with only nonnegative coefficients? 
  
Given n, let a(n) be the largest exponent k such that: 
   G(x) = 1 + x*[d/dx x*G(x)^n]/G(x)^k 
is satisfied by a power series in x with positive coefficients.  
  
Then it appears that the ratio of a(n)/n, as n grows, approaches a limit:   
 
   Limit a(n)/n = 1.541904116820940344888204990518... 
  
Below are some examples of how a(n) grows for n=10^m. 
  
Please verify; I only checked the initial 31 coefficients for negative values, so all this needs validation. 
 
Two requests: could someone 
(1) generate the sequence a(n) for n=1,2,3,...
(2) generate more digits of the above constant? 
 
Thanks, 
   Paul  
n     a(n)  
10^0  1
10^1  14
10^2  153
10^3  1541
10^4  15418 
10^5  154189
10^6  1541903
10^7  15419040
10^8  154190411
10^9  1541904116
10^10 15419041167
10^11 154190411681
10^12 1541904116820
10^13 15419041168208
10^14 154190411682093
10^15 1541904116820939
10^16 15419041168209402
10^17 154190411682094033
10^18 1541904116820940344
10^19 15419041168209403448
10^20 154190411682094034488
10^21 1541904116820940344887
10^22 15419041168209403448881 
10^23 154190411682094034488819
10^24 1541904116820940344888204
10^25 15419041168209403448882049
10^26 154190411682094034488820498
10^27 1541904116820940344888204990
10^28 15419041168209403448882049904
10^29 154190411682094034488820499051
10^30 1541904116820940344888204990517
10^31 15419041168209403448882049905179
10^32 154190411682094034488820499051803 
...