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%I #13 May 28 2023 10:43:30
%S 1,61,1441,17861,142001,819005,3713305,13980205,45432805,131091505,
%T 342981013,826861993,1859914733,3942293993,7937011013,15276834025,
%U 28261896025,50477521525,87368496025,147013666525,241153442041,386532523301,606631094081,933869816501
%N Crystal ball sequence for the A_5 x A_5 lattice.
%C 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.
%H T. D. Noe, <a href="/A143011/b143011.txt">Table of n, a(n) for n = 0..1000</a>
%H R. Bacher, P. de la Harpe and B. Venkov, <a href="http://archive.numdam.org/ARCHIVE/AIF/AIF_1999__49_3/AIF_1999__49_3_727_0/AIF_1999__49_3_727_0.pdf">Series de croissance et series d'Ehrhart associees aux reseaux de racines</a>, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).
%F 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)).
%F 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]
%p 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);
%Y Cf. A143007, A143008, A143009, A143010.
%K easy,nonn
%O 0,2
%A _Peter Bala_, Jul 22 2008
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