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A008397 Coordination sequence for E_7 lattice.

%I A008397 #38 May 30 2023 14:11:26
%S A008397 1,126,2898,25886,133506,490014,1433810,3573054,7902594,15942206,
%T A008397 29896146,52834014,88892930,143501022,223622226,338022398,497556738,
%U A008397 715478526,1007769170,1393489566
%N A008397 Coordination sequence for E_7 lattice.
%H A008397 T. D. Noe, <a href="/A008397/b008397.txt">Table of n, a(n) for n = 0..1000</a>
%H A008397 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 A008397 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 A008397 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 A008397 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A008397 a(n) = (2/5)*(74*n^6 - 6*n^5 + 130*n^4 + 30*n^3 + 106*n^2 - 24*n + 5) for n >= 1.
%F A008397 Bacher et al. give a g.f.
%F A008397 G.f.: (1 + 119*x + 2037*x^2 + 8211*x^3 + 8787*x^4 + 2037*x^5 + 119*x^6 + x^7)/(1-x)^7. - _Colin Barker_, Sep 26 2012
%p A008397 a:= n-> `if`(n=0, 1, 148/5*n^6-12/5*n^5+52*n^4+12*n^3+212/5*n^2-48/5*n+2):
%p A008397 seq(a(n), n=0..25);
%t A008397 LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,126,2898,25886,133506, 490014,1433810,3573054},20] (* _Harvey P. Dale_, Nov 12 2014 *)
%o A008397 (Magma) [1] cat [(2/5)*(74*n^6 -6*n^5 +130*n^4 +30*n^3 +106*n^2 -24*n + 5): n in [1..30]]; // _G. C. Greubel_, May 29 2023
%o A008397 (SageMath) [2*(74*n^6 -6*n^5 +130*n^4 +30*n^3 +106*n^2 -24*n +5)//5 - int(n==0) for n in range(31)] # _G. C. Greubel_, May 29 2023
%K A008397 nonn,easy
%O A008397 0,2
%A A008397 _N. J. A. Sloane_ and _J. H. Conway_