Sequences A120588 - A120607 : Formulas?

[Paul D. Hanna]

Seqfans, 
      Sequences  A120588 - A120607  are examples of 
solutions to functional equations of the form: 
    r*A(x) = c + b*x + A(x)^n 
I list the examples at the bottom of this message. 
 
Can anyone find nice formulas for any of these sequences? 
A120588 is simple, but what about A120590 - A120607 ? 
 
In A120588 I give a comment describing the conditions necessary 
to satisfy such an equation, but the comment needs to be revised 
and made more precise.  Would someone care to help? 
 
Thanks, 
      Paul 
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A120588 - COMMENT :
 
"In order for the g.f. of an integer sequence to satisfy 
a functional equation of the form: 
  r*A(x) = c + b*x + A(x)^n, where n > 1, 
it is necessary that the sequence start with [1, d, m*n*(n-1)/2], 
where d divides m*n*(n-1)/2 (m>0), and that 
the coefficients are given by r = n + d^2/m, c = r-1, and b = d^3/m. 

The remaining terms may then be integer and still satisfy: 
  a_n(k) = r*a(k), 
where a_n(k) is the k-th term of the n-th self-convolution of the
sequence."
 
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EXAMPLES: A-numbers, g.f. equation, initial 3 terms. 
 
A120588 : 3*A(x) = 2 + x + A(x)^2       : [1,1,1].
A120590 : 4*A(x) = 3 + x + A(x)^3       : [1,1,3]. 
A120592 : 5*A(x) = 4 + 4*x + A(x)^3     : [1,2,6].
A120593 : 5*A(x) = 4 + x + A(x)^4       : [1,1,6].
A120594 : 8*A(x) = 7 + 8*x + A(x)^4     : [1,2,6].
A120595 : 13*A(x) = 12 + 27*x + A(x)^4  : [1,3,6].
A120596 : 6*A(x) = 5 + x + A(x)^5       : [1,1,10].
A120597 : 9*A(x) = 8 + 8*x + A(x)^5     : [1,2,10].
A120598 : 30*A(x) = 29 + 125*x + A(x)^5 : [1,5,10]. 
A120599 : 13*A(x) = 12 + 32*x + A(x)^5  : [1,4,20].
A120600 : 7*A(x) = 6 + x + A(x)^6       : [1,1,15].
A120601 : 15*A(x) = 14 + 27*x + A(x)^6  : [1,3,15]. 
A120602 : 31*A(x) = 30 + 125*x + A(x)^6 : [1,5,15]. 
A120603 : 16*A(x) = 15 + 27*x + A(x)^7  : [1,3,21].  
A120604 : 24*A(x) = 23 + 64*x + A(x)^8  : [1,4,28].  
A120605 : 25*A(x) = 24 + 64*x + A(x)^9  : [1,4,36].  
A120606 : 36*A(x) = 35 + 81*x + A(x)^9  : [1,3,12].  
A120607 : 37*A(x) = 36 + 81*x + A(x)^10 : [1,3,15].  
 
END.