More Functional Puzzles

[Paul D. Hanna]

Seqfans, 
     Here are 3 more (compare to (2a) and (2b) below).  
There seems to be a pattern, but I do not have it all figured out yet. 
Note that in each of these pairs of problems, such as (4a) and (4b), 
only A is the same; solutions for B,C,D,... are different for each
problem. 
    Paul 
-------------------------------------------------------------------
Consider the systems of simultaneous equations below; for each case, 
what is the unique solution to variable A as a formal power series in x? 

(4a)
A = 1 + xB 
B = A + xD
C = B + xF
D = C + xH
E = D + xJ
...
(4b)
A = 1 + xAB
B = 1 + xBCD
C = 1 + xCDEF
D = 1 + xDEFGH
E = 1 + xEFGHIJ
...
(same solution for A as in (4a)).
 
(5a)
A =  (1 + xAB)
B = A(1 + xBC)
C = B(1 + xCD)
D = C(1 + xDE)
E = D(1 + xEF)
...
(5b)  
A = 1 + x(A)^2*B
B = 1 + x(AB)^2*C
C = 1 + x(ABC)^2*D
D = 1 + x(ABCD)^2*E
E = 1 + x(ABCDE)^2*F 
...
(same solution for A as in (5a)).
 
(6a)
A =  (1 + xC)
B = A(1 + xD)
C = B(1 + xE)
D = C(1 + xF)
E = D(1 + xG)
...
(6b)  
A = 1 + xABC
B = 1 + xABCD
C = 1 + xABCDE
D = 1 + xABCDEF
E = 1 + xABCDEFG 
...
(same solution for A as in (6a)).

 
SOLUTIONS. 
(4a) and (4b):
A = g.f. of A002449 
(number of different types of binary trees of height n): 
[1, 1, 2, 6, 26, 166, 1626, 25510, 664666, 29559718, 2290267226,...]. 
  
(5a) and (5b): 
A satisfies:  A(x) = 1 + xA(x)^2*A(xA(x)^2) = g.f. of A088717:
[1, 1, 3, 14, 84, 596, 4785, 42349, 406287, 4176971, 45640572,...]. 
 
(6a) and (6b): 
x*A = A(x) satisfies:  A(x) = x + x*A(A(A(x))) = g.f. of A091713, 
and coefficients of A begin:
[1, 1, 3, 15, 99, 781, 7001, 69253, 742071, 8506775, 103411463,...].
  
END.
  
On Mon, 19 Mar 2007 22:55:03 -0400 "Paul D. Hanna" <pauldhanna at juno.com>
writes:
Consider the systems of simultaneous equations below; for each case, 
what is the unique solution to variable A as a formal power series in x? 
 
(1)
A = 1 + xAB
B = A + xBC
C = B + xCD
D = C + xDE
E = D + xEF
...
   
(2a) 
A = (1 + xB)
B = A(1 + xC)
C = B(1 + xD)
D = C(1 + xE)
E = D(1 + xF)
... 
(2b)  
A = 1 + xAB
B = 1 + xABC
C = 1 + xABCD
D = 1 + xABCDE
E = 1 + xABCDEF 
...
(same solution as (2a)).
 
(3) 
A = 1 + xB
B = 1 + xAC
C = 1 + xABD
D = 1 + xABCE
E = 1 + xABCDF
...
     
SOLUTIONS.
(1) A satisfies:  A(x) = 1 + x*A(x)^2*A(xA(x)) = g.f. of A088714: 
[1, 1, 3, 13, 69, 419, 2809, 20353, 157199, 1281993, 10963825,...].  
  
(2a) and (2b): A satisfies:  A(x) = 1 + xA(x)*A(xA(x)) 
where xA(x) = g.f. of A030266, and coefficients of A begin: 
[1, 1, 2, 6, 23, 104, 531, 2982, 18109, 117545, 808764,...]. 
 
(3) A satisfies:  A(x) = 1 + xA(xA(x)) = g.f. of A087949: 
[1, 1, 1, 2, 5, 16, 59, 246, 1131, 5655, 30428, 174835,...]. 

END.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://list.seqfan.eu/pipermail/seqfan/attachments/20070320/6277c8c1/attachment-0001.htm>