A035839 Coordination sequence for A_13 lattice.
1, 182, 8372, 176722, 2206932, 18827718, 120353324, 614266354, 2619716554, 9654482474, 31534801116, 93093230958, 252208679268, 634756203018, 1498750896708, 3346707628446, 7114703302434, 14479567043214, 28342922553764, 53573492643034, 98118326104708
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
- J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
- Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
- Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).
Programs
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Mathematica
CoefficientList[Series[-(x + 1) (x^12 + 168 x^11 + 5916 x^10 + 75880 x^9 + 435345 x^8 + 1221024 x^7 + 1723632 x^6 + 1221024 x^5 + 435345 x^4 + 75880 x^3 + 5916 x^2 + 168 x + 1)/(x - 1)^13, {x, 0, 30}], x] (* Vincenzo Librandi, Oct 21 2013 *)
Formula
Sum_{d=1..13} C(14, d)*C(m/2-1, d-1)*C(13-d+m/2, m/2), where norm m is always even.
G.f.: -(x+1)*(x^12 + 168*x^11 + 5916*x^10 + 75880*x^9 + 435345*x^8 + 1221024*x^7 + 1723632*x^6 + 1221024*x^5 + 435345*x^4 + 75880*x^3 + 5916*x^2 + 168*x + 1) / (x-1)^13. [Colin Barker, Nov 19 2012]
Extensions
More terms from Vincenzo Librandi, Oct 21 2013