[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]
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