cp's OEIS Frontend

A143011 Crystal ball sequence for the A_5 x A_5 lattice.

%I A143011 #18 Aug 21 2025 16:54:27
%S A143011 1,61,1441,17861,142001,819005,3713305,13980205,45432805,131091505,
%T A143011 342981013,826861993,1859914733,3942293993,7937011013,15276834025,
%U A143011 28261896025,50477521525,87368496025,147013666525,241153442041,386532523301,606631094081,933869816501
%N A143011 Crystal ball sequence for the A_5 x A_5 lattice.
%C A143011 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 A143011 T. D. Noe, <a href="/A143011/b143011.txt">Table of n, a(n) for n = 0..1000</a>
%H A143011 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">Séries de croissance et séries d'Ehrhart associées aux réseaux de racines</a>, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
%H A143011 <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 A143011 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.
%F A143011 O.g.f.: 1/(1-x)*[Legendre_P(5,(1+x)/(1-x))]^2.
%F A143011 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)).
%F A143011 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 A143011 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);
%t A143011 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;
%t A143011 Array[A143011, 50, 0] (* _Paolo Xausa_, Aug 21 2025 *)
%Y A143011 Row 5 of A143007.
%Y A143011 Cf. A143008, A143009, A143010.
%K A143011 easy,nonn
%O A143011 0,2
%A A143011 _Peter Bala_, Jul 22 2008