[seqfan] Re: Zeros in A172390 and A172391

[Paul D Hanna]

Seqfans, 
      Is it known/trivial that: 
(5) [x^(4n+2)] EllipticK(4x)^(-4n) = 0 for n>=1. 
 
This statement (5) is equivalent to (4) given in prior email. 
 
If (5) is true, it would imply that   A172390(2n+1) = 0  for n>=1. 
 
Thanks, 
     Paul 
---------- Original Message ----------
From: "Paul D Hanna" <pauldhanna at juno.com>
To: seqfan at list.seqfan.eu
Subject: [seqfan]  Zeros in A172390 and A172391
Date: Sat, 20 Mar 2010 16:29:42 GMT

SeqFans, 
    Sequences A172390 and A172391 record 2 surprising observations. 
Is there any reason why the following statements should be true? 
(1)  A172390(2n+1) = 0  for n>=1; 
(2)  A172391(2n+1) = 0  for n>=1. 
 
Here is a fact that may be a big clue for (1):  
(3)  Sum_{n>=0} C(2n,n)^2*x^n  =  1/AGM(1, (1-16x)^(1/2) )
where AGM is the arithmetic-geometric mean. 
 
Statement (3) makes (1) equivalent to: 
(4)  [x^(2n+1)] AGM(1, (1-16x)^(1/2) )^(4n)  = 0 for n>=1. 

Can someone show that this (4) is true? 
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