A143011 Crystal ball sequence for the A_5 x A_5 lattice.

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

Author

Peter Bala, Jul 22 2008

Keywords

• easy
• nonn
• changed

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

• T. D. Noe, Table of n, a(n) for n = 0..1000
• R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux 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).

Crossrefs

Row 5 of A143007.

Cf. A143008, A143009, A143010.

Programs

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);

Mathematica

A143011[n_] := (n*(n + 1)*(7*n*(n + 1)*(n*(n + 1)*(9*n*(n + 1)*(2*n*(n + 1) + 45) + 2644) + 6028) + 32880))/7200 + 1;
Array[A143011, 50, 0] (* Paolo Xausa, Aug 21 2025 *)

Formula

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..oo} 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]