login
Coordination sequence for C_9 lattice.
6

%I #24 Feb 03 2023 01:40:33

%S 1,162,4482,53154,374274,1854882,7159170,22952610,63821826,158611106,

%T 360027522,758497698,1501390338,2818849698,5057616258,8724341922,

%U 14540038146,23507426466,36993091970,56826471330,85417838082,125897578914,182279185794,259648519842

%N Coordination sequence for C_9 lattice.

%H Seiichi Manyama, <a href="/A035746/b035746.txt">Table of n, a(n) for n = 0..10000</a>

%H R. Bacher, P. de la Harpe and B. Venkov, <a href="https://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 Joan Serra-Sagrista, <a href="https://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_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).

%F a(n) = [x^(2n)] ((1+x)/(1-x))^9.

%F a(n) = A008418(2*n). - _Seiichi Manyama_, Jun 08 2018

%F From _Chai Wah Wu_, Feb 02 2023: (Start)

%F a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n > 9.

%F G.f.: -(x + 1)*(x^2 + 14*x + 1)*(x^6 + 138*x^5 + 975*x^4 + 1868*x^3 + 975*x^2 + 138*x + 1)/(x - 1)^9. (End)

%Y Cf. A008418.

%K nonn,easy

%O 0,2

%A Joan Serra-Sagrista (jserra(AT)ccd.uab.es)

%E Recomputed by _N. J. A. Sloane_, Nov 25 1998