[seqfan] Re: 1,6,110,2562,66222,...

[Edwin Clark]

I finally took the effort to see that the sequence really is a subsequence 
of the array S(d,n) = coordination sequence of the lattice A_d: If anyone 
is interested in the details see:

%I A156554
%S A156554 1,6,110,2562,66222,1815506,51697802,1511679210,45076309166,
%T A156554 
1364497268946,41800229045610,1292986222651646,40317756506959050,
%U A156554 1265712901796074842,39965073938276694002,1268208750951634765562
%N A156554 The number of integer sequences of length d = 2n+1 such that 
the sum of the terms is 0
and the sum of the absolute values of the terms is d-1.
%C A156554 Let b(n) = S(d,n) be the coordination sequence of the lattice 
A_d. Then this sequence is
a(n) = S(2n,n). See [1]. The sequence is defined in [2].
%H A156554 [1] J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices 
VII: Coordination
Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a
href="http://www.research.att.com/~njas/doc/ldl7.txt">Abstract</a>, 
<a
href="http://www.research.att.com/~njas/doc/ldl7.pdf">pdf</a>, 
<a
href="http://www.research.att.com/~njas/doc/ldl7.ps">ps</a>).
%H A156554 [2] J.-M. Couveignes, T. Ezome and R. Lercier. Elliptic periods 
and primality proving,
(2008) (<a 
href="http://www.math.univ-toulouse.fr/~couveig/publi/arxiv3.pdf">pdf</a>.
%F A156554 a(n) = S(2n,n) where S(d,n) = Sum(k=0..d,C(d,k)^2*C(n-k+d-1)) 
from formula (22) in [1].
%e A156554 For n = 1 the a(n) = 6 sequences are
(1,-1,0),(-1,1,0),(1,0,-1),(-1,0,1),(0,1,-1),(0,-1,1)
%p A156554 S:=proc(d,n) add(binomial(d,k)^2*binomial(n-k+d-1,d-1),k=0..d); 
end proc;
a:=n->S(2*n,n);
%Y A156554 a(n)=A103881(2n,n)
%K A156554 easy,nice,nonn
%O A156554 0,2
%A A156554 W. Edwin Clark (eclark(AT)math.usf.edu), Feb 09 2009