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A142994 Crystal ball sequence for the lattice C_5.

%I A142994 #41 Mar 23 2024 08:06:06
%S A142994 1,51,501,2471,8361,22363,50973,103503,192593,334723,550725,866295,
%T A142994 1312505,1926315,2751085,3837087,5242017,7031507,9279637,12069447,
%U A142994 15493449,19654139,24664509,30648559,37741809,46091811,55858661,67215511,80349081
%N A142994 Crystal ball sequence for the lattice C_5.
%C A142994 The lattice C_5 consists of all integer lattice points v = (x_1,...,x_5) in Z^5 such that (x_1 + ... + x_5) is even, equipped with the taxicab type norm ||v|| = (1/2) * (|x_1| + ... + |x_5|). The crystal ball sequence of C_5 gives the number of lattice points v in C_5 with ||v|| <= n for n = 0,1,2,3,... [Bacher et al.].
%C A142994 Partial sums of A019561.
%H A142994 Seiichi Manyama, <a href="/A142994/b142994.txt">Table of n, a(n) for n = 0..10000</a>
%H A142994 R. Bacher, P. de la Harpe, and B. Venkov, <a href="https://doi.org/10.5802/aif.1689">Séries de croissance et séries d'Ehrhart associées aux réseaux de racines</a>, Annales de l'Institut Fourier, Tome 49 (1999) no. 3, pp. 727-762.
%H A142994 R. Bacher, P. de la Harpe and B. Venkov, <a href="https://doi.org/10.1016/S0764-4442(97)83542-2">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 A142994 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A142994 a(n) = (2*n + 1)*(32*n^4 + 64*n^3 + 88*n^2 + 56*n + 15)/15.
%F A142994 a(n) = Sum_{k = 0..5} binomial(10, 2*k)*binomial(n+k, 5).
%F A142994 a(n) = Sum_{k = 0..5} binomial(10, 2*k+1)*binomial(n+k+1/2, 5).
%F A142994 O.g.f.: (1 + 45*x + 210*x^2 + 210*x^3 + 45*x^4 + x^5)/(1 - x)^6 = 1/(1 - x) * T(5, (1 + x)/(1 - x)), where T(n, x) denotes the Chebyshev polynomial of the first kind.
%F A142994 Sum_{n >= 1} 1/(n*a(n-1)*a(n)) = 2*log(2) - 41/30.
%F A142994 a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6), for n > 5. - _Vincenzo Librandi_, Dec 16 2015
%F A142994 From _Peter Bala_, Mar 11 2024: (Start)
%F A142994 Sum_{k = 1..n+1} 1/(k*a(k)*a(k-1)) = 1/(51 - 3/(59 - 60/(75 - 315/(99 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*5^2))))).
%F A142994 E.g.f.: exp(x)*(1 + 50*x + 400*x^2/2! + 1120*x^3/3! + 1280*x^4/4! + 512*x^5/5!).
%F A142994 Note that -T(10, i*sqrt(x)) = 1 + 50*x + 400*x^2 + 1120*x^3 + 1280*x^4 + 512*x^5. See A008310. (End)
%e A142994 a(1) = 51. The origin has norm 0. The 50 lattice points in Z^5 of norm 1 (as defined above) are +-2*e_i, 1 <= i <= 5 and (+- e_i +- e_j), 1 <= i < j <= 5, where e_1, ... , e_5 denotes the standard basis of Z^5. These 50 vectors form a root system of type C_5. Hence the sequence begins 1, 1 + 50 = 51, ... .
%p A142994 a := n -> (2*n+1)*(32*n^4+64*n^3+88*n^2+56*n+15)/15: seq(a(n), n = 0..20)
%t A142994 CoefficientList[Series[(1 + 45 x + 210 x^2 + 210 x^3 + 45 x^4 + x^5)/(1 - x)^6, {x, 0, 33}], x] (* or *) LinearRecurrence[{6, -15, 20, -15, 6, -1},{1, 51, 501, 2471, 8361, 22363}, 25] (* _Vincenzo Librandi_, Dec 16 2015 *)
%o A142994 (Python)
%o A142994 A142994_list, m = [], [512, -768, 352, -48, 2, 1]
%o A142994 for _ in range(10**2):
%o A142994     A142994_list.append(m[-1])
%o A142994     for i in range(5):
%o A142994         m[i+1] += m[i] # _Chai Wah Wu_, Dec 15 2015
%o A142994 (Magma) [(2*n+1)*(32*n^4+64*n^3+88*n^2+56*n+15)/15: n in [0..30]]; // _Vincenzo Librandi_, Dec 16 2015
%Y A142994 Row 5 of A142992. Cf. A019561, A063496, A142993.
%K A142994 nonn,easy
%O A142994 0,2
%A A142994 _Peter Bala_, Jul 18 2008