# [seqfan] Re: Zeros in A172390 and A172391

Paul D Hanna pauldhanna at juno.com
Mon Mar 22 18:24:45 CET 2010

```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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