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%I #22 Sep 08 2022 08:45:37
%S 1,41,661,5741,33001,142001,494341,1465661,3833941,9073501,19789001,
%T 40328641,77620661,142282141,250054001,423621001,694880441,1107728161,
%U 1721435341,2614694501,3890418001,5681377241,8156775661,11529853541
%N Crystal ball sequence for the A4 x A4 lattice.
%C 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.
%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">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 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36, 84,-126,126,-84,36,-9,1).
%F 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.
%F O.g.f. : 1/(1-x)*[Legendre_P(4,(1+x)/(1-x))]^2.
%F 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)).
%F G.f.: (1+16*x+36*x^2+16*x^3+x^4)^2/(1-x)^9. [_Colin Barker_, Mar 16 2012]
%F 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 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);
%t 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 *)
%o (Python)
%o A143010_list, m = [], [4900, -14700, 17500, -10500, 3340, -540, 40, 0, 1]
%o for _ in range(10**2):
%o A143010_list.append(m[-1])
%o for i in range(8):
%o m[i+1] += m[i] # _Chai Wah Wu_, Dec 15 2015
%o (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
%Y Cf. A143007 (row 4), A143008, A143009, A143011.
%K easy,nonn
%O 0,2
%A _Peter Bala_, Jul 22 2008
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