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%I #35 Dec 10 2023 16:02:58
%S 1,240,9120,121680,864960,4113840,14905440,44480400,114879360,
%T 265422960,561403680,1105317840,2050966080,3620750640,6126497760,
%U 9994133520,15792541440,24266930160,36377039520,53340513360,76681767360
%N Coordination sequence for E_8 lattice.
%D M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
%H T. D. Noe, <a href="/A008340/b008340.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, Zeit. f. Kristallographie, 212 (1997), 253-256.
%H R. Bacher, P. de la Harpe and B. Venkov, <a href="http://dx.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 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E8.html">Home page for this lattice</a>
%H M. O'Keeffe, <a href="http://dx.doi.org/10.1524/zkri.1995.210.12.905">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908.
%H M. O'Keeffe, <a href="/A008527/a008527.pdf">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
%H <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 a(n) = if n = 0 then 1 else (456/7)*n^7-120*n^6+312*n^5-120*n^4-48*n^3+240*n^2-(624/7)*n.
%F Bacher et al. give a g.f.
%F G.f.: (x^8 +232*x^7 +24508*x^6 +107224*x^5 +133510*x^4 +55384*x^3 +7228*x^2 +232*x +1)/(x -1)^8 = 1 + 240*x* (1+30*x+231*x^2+556*x^3+447*x^4+102*x^5+x^6) /(1-x)^8. [_Colin Barker_, Sep 26 2012]
%p if n = 0 then 1 else 456/7*n^7-120*n^6+312*n^5-120*n^4-48*n^3+240*n^2-624/7*n;
%t Join[{1},Table[456/7*n^7-120*n^6+312*n^5-120*n^4-48*n^3+ 240*n^2- 624/7*n,{n,20}]] (* _Harvey P. Dale_, Jul 14 2014 *)
%Y Cf. A019557, A019558, A008397, A008399.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_ and _J. H. Conway_
%E The values given by O'Keeffe are incorrect.
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