0,2

The A_4 lattice consists of all vectors v = (a,b,c,d,e) in Z^5 such that a+b+c+d+e = 0. The lattice is equipped with the norm ||v|| = 1/2*(|a| + |b| + |c| + |d| + |e|). Pairs of lattice points (v,w) in the product lattice A_4 x A_4 have norm ||(v,w)|| = ||v|| + ||w||. Then the k-th term in the crystal ball sequence for the A_4 x A_4 lattice gives the number of such pairs (v,w) for which ||(v,w)|| is less than or equal to k.

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 (9,-36, 84,-126,126,-84,36,-9,1).

a(n) = (35*n^8 +140*n^7 +630*n^6 +1400*n^5 +2595*n^4 +3020*n^3 +2500*n^2 +1200*n +288)/288 = 5*n*(n + 1)*(n^2 + n + 2)*(7*n^4 + 14*n^3 + 77*n^2 + 70*n + 120)/288 + 1.

O.g.f. : 1/(1-x)*[Legendre_P(4,(1+x)/(1-x))]^2.

Apery's constant zeta(3) = (1+1/2^3+1/3^3+1/4^3) + Sum {n = 1..inf} 1/(n^3*a(n-1)*a(n)).

G.f.: (1+16*x+36*x^2+16*x^3+x^4)^2/(1-x)^9. [Colin Barker, Mar 16 2012]

a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>8. - Vincenzo Librandi, Dec 16 2015

p := n -> (35*n^8 +140*n^7 +630*n^6 +1400*n^5 +2595*n^4 +3020*n^3 +2500*n^2 +1200*n +288)/288: seq(p(n), n = 0..24);

LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 41, 661, 5741, 33001, 142001, 494341, 1465661, 3833941}, 25] (* Vincenzo Librandi, Dec 16 2015 *)

(Python)

A143010_list, m = [], [4900, -14700, 17500, -10500, 3340, -540, 40, 0, 1]

for _ in range(10**2):

A143010_list.append(m[-1])

for i in range(8):

m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015

(Magma) [5*n*(n+1)*(n^2+n+2)*(7*n^4+14*n^3+77*n^2+70*n+120)/288+1: n in [0..30]]; // Vincenzo Librandi, Dec 16 2015

Cf. A143007 (row 4), A143008, A143009, A143011.

Sequence in context: A299600 A197371 A268748 * A009730 A009761 A118448

Adjacent sequences: A143007 A143008 A143009 * A143011 A143012 A143013

easy,nonn

Peter Bala, Jul 22 2008

approved