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Coordination sequence for A_10 lattice.
3

%I #40 May 28 2023 04:04:47

%S 1,110,3080,40370,322190,1815506,7925720,28512110,88206140,241925530,

%T 601585512,1379301990,2953859370,5968878630,11472968760,21114177018,

%U 37403270520,64062783510,106481351240,172295622730,272125000774,420487598410

%N Coordination sequence for A_10 lattice.

%H T. D. Noe, <a href="/A008395/b008395.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Baake and U. Grimm, <a href="https://arxiv.org/abs/cond-mat/9706122">Coordination sequences for root lattices and related graphs</a>, arXiv:cond-mat/9706122, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.

%H 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 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).

%H Joan Serra-Sagrista, <a href="http://dx.doi.org/10.1016/S0020-0190(00)00119-8">Enumeration of lattice points in l_1 norm</a>, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

%F a(n) = 46189/90720*n^9 +26741/3024*n^7 +171457/4320*n^5 +111683/2268*n^3 +7381/630*n for n >= 1.

%F Sum_{d=1}^10 C(11, d) C(m/2-1, d-1) C(10-d+m/2, m/2), where norm m is always even. (Serra-Sagrista)

%F G.f.: (1 +100*x +2025*x^2 +14400*x^3 +44100*x^4 +63504*x^5 +44100*x^6 +14400*x^7 +2025*x^8 +100*x^9 +x^10)/(1-x)^10. - _Colin Barker_, Sep 26 2012

%p a:= n-> `if`(n=0, 1, 46189/90720*n^9+26741/3024*n^7+

%p 171457/4320*n^5+111683/2268*n^3+7381/630*n):

%p seq(a(n), n=0..25);

%t a[n_]:= If[n==0, 1, 11n(4199n^8 +72930n^6 +327327n^4 +406120n^2 +96624)/90720];

%t Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Jan 07 2019 *)

%t LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1}, {1,110,3080, 40370,322190,1815506,7925720,28512110,88206140,241925530,601585512}, 30] (* _Harvey P. Dale_, Nov 27 2019 *)

%o (Magma) [1] cat [11*n*(4199*n^8 +72930*n^6 +327327*n^4 +406120*n^2 +96624)/90720: n in [1..40]]; // _G. C. Greubel_, May 27 2023

%o (SageMath) [11*n*(4199*n^8 +72930*n^6 +327327*n^4 +406120*n^2 +96624)//90720 +int(n==0) for n in range(41)] # _G. C. Greubel_, May 27 2023

%Y Row 10 of A103881.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_ and _J. H. Conway_