[seqfan] Re: Zeros in A172390 and A172391

[Paul D Hanna]

SeqFans, 
     Just realized that the statements in the prior emails reduce to the following. 
Let 
  A(x) = g.f. of A172390, and
  B(x) = g.f. of A172391;
then 
  A(x)^(1/2) = F(x^2) + 4x/F(x^2) and
  B(x)^(1/2) = G(x^2) + 4x/G(x^2). 
 
This statement seems significant ... what does it imply about A172390 and A172391? 
 
All the statements in the prior emails are simple deductions from the above - 
so sorry for the email clutter!!  
     Paul 
 
---------- Original Message ----------
From: "Paul D Hanna" <pauldhanna at juno.com>
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: Zeros in A172390 and A172391
Date: Mon, 29 Mar 2010 21:06:44 GMT

SeqFans, 
     To expand on Joerg's observation, 
here is a relation that both sequences A172390 and A172391 seem to have in common. 
No doubt the relation is due to their elliptic connections. 

Let 
 A(x) = g.f. of A172390, and
 B(x) = g.f. of A172391;

then the following functions equal the sum of the respective quadrasections defined by: 
 sqrt( sqrt(A(x^2)) + 4*x ) = QA0(x^4) + x*QA1(x^4) 
 sqrt( sqrt(B(x^2)) + 4*x ) = QB0(x^4) + x*QB1(x^4) 

such that the quadrasections satisfy the product: 
 QA0(x^4)*QA1(x^4)  =  QB0(x^4)*QB1(x^4)  =  2 
and also 
 QA0(x^2)^2 + x*QA1(x^2)^2 = sqrt(A(x)) ;
 QB0(x^2)^2 + x*QB1(x^2)^2 = sqrt(B(x)).

Please see examples below.  

Is this relation trivial? 
  Paul 
----------------------------------------------------------------
[...]