login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A143011
Crystal ball sequence for the A_5 x A_5 lattice.
5
1, 61, 1441, 17861, 142001, 819005, 3713305, 13980205, 45432805, 131091505, 342981013, 826861993, 1859914733, 3942293993, 7937011013, 15276834025, 28261896025, 50477521525, 87368496025, 147013666525, 241153442041, 386532523301, 606631094081, 933869816501
OFFSET
0,2
COMMENTS
The A_5 lattice consists of all vectors v = (x_1,...,x_6) in Z^6 such that sum {i = 1..6} x_i = 0. The lattice is equipped with the norm ||v|| = 1/2*(sum {i = 1..6} |x_i|). Pairs of lattice points (v,w) in the product lattice A_5 x A_5 have norm ||(v,w)|| = ||v|| + ||w||. Then the k-th term in the crystal ball sequence for the A_5 x A_5 lattice gives the number of such pairs (v,w) for which ||(v,w)|| is less than or equal to k.
LINKS
R. Bacher, P. de la Harpe and B. Venkov, Series de croissance et series d'Ehrhart associees aux reseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).
FORMULA
Row 5 of A143007. a(n) := (126*n^10 +630*n^9 +4095*n^8 +12600*n^7 +36148*n^6 +66990*n^5 +100555*n^4 +102900*n^3 +75076*n^2 +32880*n +7200)/7200. O.g.f. : 1/(1-x)*[Legendre_P(5,(1+x)/(1-x))]^2. Apery's constant zeta(3) = (1+1/2^3+1/3^3+1/4^3+1/5^3) + sum {n = 1..inf} 1/(n^3*a(n-1)*a(n)).
G.f.: (1+x)^2*(1+24*x+76*x^2+24*x^3+x^4)^2/(1-x)^11. [Colin Barker, Apr 16 2012]
MAPLE
p := n -> (126*n^10 +630*n^9 +4095*n^8 +12600*n^7 +36148*n^6 +66990*n^5 +100555*n^4 +102900*n^3 +75076*n^2 +32880*n +7200)/7200: seq(p(n), n = 0..24);
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Peter Bala, Jul 22 2008
STATUS
approved