# [seqfan] Zeros in A172390 and A172391

Paul D Hanna pauldhanna at juno.com
Sat Mar 20 17:29:42 CET 2010

```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?

I don't know where to begin to show (2).

Below I copy the gist of the sequences.
Thanks,
Paul
------------------------------------------------------------------------
A172390
[1,8,24,0,-168,0,2112,0,-32040,0,536256,0,-9542976,0,177126912,0,-3390361128,0,...]

G.f. satisfies:
A(x) = G(x/A(x))^2
where
G(x) = Sum_{n>=0} C(2n,n)^2 *x^n.

(PARI)
{a(n)=local(G=sum(m=0,n,binomial(2*m,m)^2*x^m)+x*O(x^n));polcoeff(x/serreverse(x*G^2),n)}
------------------------------------------------------------------------
A172391
[1,8,12,0,28,0,264,0,3720,0,63840,0,1232432,0,25731216,0,568130552,0,...]

G.f. satisfies:
A(x) = G(x/A(x))^2
where
G(x) = Sum_{n>=0} C(2n,n)*C(2n+2,n+1)/(n+2) *x^n.

(PARI)
{a(n)=local(G=sum(m=0,n,binomial(2*m,m)*binomial(2*m+2,m+1)/(m+2)*x^m)+x*O(x^n));polcoeff(x/serreverse(x*G^2),n)}
------------------------------------------------------------------------
[END]

```