[seqfan] Re: Zeros in A172390 and A172391

[Paul D Hanna]

Joerg (and SeqFans), 
     Here is a related observation. 
 
Take the convolution square-root of A172390, which proceeds: 
[1, 4, 4, -16, -28, 176, 336, -2496, -4956, 40112, 81488, -694720,...] 
 
and compare that to A158100: 
[1,4,4,0,4,0,-16,0,-28,0,176,0,336,0,-2496,0,-4956,0,40112,0,81488,...], 
for which the g.f. satisfies: A(x) = 1/AGM(1, 1 - 8*x/A(x) ). 
 
So if we remove the odd-indexed terms [4,0,0,0,...] from A158100 we obtain A172390^(1/2). 
And the AGM appears in both g.f.s. 
  
     Paul 
  
---------- Original Message ----------
From: Joerg Arndt <arndt at jjj.de>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Zeros in A172390 and A172391
Date: Mon, 29 Mar 2010 07:38:26 +0200

* Joerg Arndt <arndt at jjj.de> [Mar 23. 2010 06:52]:
> [...]
> 
> Interesting.  I don't think this is known.
> I'll think more about this.
> 
It appears that if the 'odd' term 8*x, i.e.
A172390  1, 8, 24, 0, -168, 0, 2112, 0, -32040, 0, 536256, ...
           ^
        this one
 
is removed then the series is a 4th power of
another one that has all integer coefficients.
Whether this is just another way of putting
the conjecture in question I do not know.
> 
> [...]