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A008391 Coordination sequence for A_8 lattice.

%I A008391 #46 May 27 2023 03:15:32
%S A008391 1,72,1332,11832,66222,271224,889716,2476296,6077196,13507416,
%T A008391 27717948,53265960,96900810,168278760,280819260,452715672,708113304,
%U A008391 1078467624,1604095524,2335932504,3337508646,4687156248,6480461988,8832976488,11883194148,15795816120
%N A008391 Coordination sequence for A_8 lattice.
%H A008391 T. D. Noe, <a href="/A008391/b008391.txt">Table of n, a(n) for n = 0..1000</a>
%H A008391 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 A008391 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 A008391 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 A008391 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F A008391 a(n) = n*(715*n^6 + 6006*n^4 + 10395*n^2 + 3044)/280, a(0) = 1.
%F A008391 a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - _Harvey P. Dale_, Mar 04 2012
%F A008391 G.f.: (1 + 64*x + 784*x^2 + 3136*x^3 + 4900*x^4 + 3136*x^5 + 784*x^6 + 64*x^7 + x^8)/(1-x)^8. - _Colin Barker_, Sep 26 2012
%p A008391 1, seq(n*(715*n^6 + 6006*n^4 +10395*n^2 +3044)/280, n=1..40);
%t A008391 Join[{1},Table[143/56n^7+429/20n^5+297/8n^3+761/70n,{n,30}]] (* or *)
%t A008391 Join[{1},LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{72,1332,11832, 66222,271224,889716,2476296,6077196},30]] (* _Harvey P. Dale_, Mar 04 2012 *)
%o A008391 (Maxima)
%o A008391 a[0]:1$
%o A008391 a[1]:72$
%o A008391 a[2]:1332$
%o A008391 a[3]:11832$
%o A008391 a[4]:66222$
%o A008391 a[5]:271224$
%o A008391 a[6]:889716$
%o A008391 a[7]:2476296$
%o A008391 a[8]:6077196$
%o A008391 a[n]:=8*a[n-1]-28*a[n-2]+ 56*a[n-3]- 70*a[n-4]+56*a[n-5]-28*a[n-6]+8*a[n-7]-a[n-8];
%o A008391 makelist(a[n],n,0,30); /* _Martin Ettl_, Oct 26 2012 */
%o A008391 (Magma) [1] cat [n*(715*n^6 + 6006*n^4 +10395*n^2 +3044)/280: n in [1..40]]; // _G. C. Greubel_, May 26 2023
%o A008391 (SageMath) [n*(715*n^6 + 6006*n^4 +10395*n^2 +3044)//280 +int(n==0) for n in range(41)] # _G. C. Greubel_, May 26 2023
%Y A008391 Row 8 of A103881.
%K A008391 nonn,easy
%O A008391 0,2
%A A008391 _N. J. A. Sloane_ and _J. H. Conway_