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A008395 Coordination sequence for A_10 lattice.

%I A008395 #40 May 28 2023 04:04:47
%S A008395 1,110,3080,40370,322190,1815506,7925720,28512110,88206140,241925530,
%T A008395 601585512,1379301990,2953859370,5968878630,11472968760,21114177018,
%U A008395 37403270520,64062783510,106481351240,172295622730,272125000774,420487598410
%N A008395 Coordination sequence for A_10 lattice.
%H A008395 T. D. Noe, <a href="/A008395/b008395.txt">Table of n, a(n) for n = 0..1000</a>
%H A008395 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 A008395 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 A008395 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 A008395 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 A008395 <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 A008395 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 A008395 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 A008395 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 A008395 a:= n-> `if`(n=0, 1, 46189/90720*n^9+26741/3024*n^7+
%p A008395          171457/4320*n^5+111683/2268*n^3+7381/630*n):
%p A008395 seq(a(n), n=0..25);
%t A008395 a[n_]:= If[n==0, 1, 11n(4199n^8 +72930n^6 +327327n^4 +406120n^2 +96624)/90720];
%t A008395 Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Jan 07 2019 *)
%t A008395 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 A008395 (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 A008395 (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 A008395 Row 10 of A103881.
%K A008395 nonn,easy
%O A008395 0,2
%A A008395 _N. J. A. Sloane_ and _J. H. Conway_