A035838 Coordination sequence for A_12 lattice.
1, 156, 6162, 112268, 1219374, 9129276, 51697802, 235895244, 907129236, 3037849828, 9079799742, 24680519604, 61908797418, 144977296932, 319917948246, 670283877588, 1341750437352, 2579499722124, 4783532975546, 8588601364668, 14977318285254, 25437258929836
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 (12, -66, 220, -495, 792, -924, 792, -495, 220, -66, 12, -1).
Programs
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Mathematica
CoefficientList[Series[(x^12 + 144 x^11 + 4356 x^10 + 48400 x^9 + 245025 x^8 + 627264 x^7 + 853776 x^6 + 627264 x^5 + 245025 x^4 + 48400 x^3 + 4356 x^2 + 144 x + 1)/(x - 1)^12, {x, 0, 30}], x] (* Vincenzo Librandi, Oct 21 2013 *)
Formula
Sum_{d=1..12} C(13, d)*C(m/2 - 1, d - 1)*C(12 - d + m/2, m/2), where norm m is always even.
G.f.: (x^12 + 144*x^11 + 4356*x^10 + 48400*x^9 + 245025*x^8 + 627264*x^7 + 853776*x^6 + 627264*x^5 + 245025*x^4 + 48400*x^3 + 4356*x^2 + 144*x + 1) / (x - 1)^12. [Colin Barker, Nov 19 2012]
Extensions
More terms from Vincenzo Librandi, Oct 21 2013