A022148 Coordination sequence for root lattice B_6.

1, 72, 1072, 6968, 28320, 85992, 214864, 467544, 918080, 1665672, 2838384, 4596856, 7138016, 10698792, 15559824, 22049176, 30546048, 41484488, 55357104, 72718776, 94190368, 120462440, 152298960

Offset: 0

Author

mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de (Michael Baake)

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), 253-256.
• 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.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Programs

Magma

[1] cat[(8*n/15)*(58*n^4 - 65*n^3 + 180*n^2 - 85*n + 47) : n in [1..40]]; // Vincenzo Librandi, Apr 20 2012

Mathematica

CoefficientList[Series[(1+66*x+655*x^2+1596*x^3+1167*x^4+ 226*x^5+x^6)/(1-x)^6,{x,0,40}],x] (* Vincenzo Librandi, Apr 20 2012 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{1,72,1072,6968,28320,85992,214864},30] (* Harvey P. Dale, Aug 11 2025 *)

Formula

a(0) = 1; for n>0, a(n) = (8*n/15)*(58*n^4 - 65*n^3 + 180*n^2 - 85*n + 47) . - Philippe Deléham, Feb 20 2004

G.f.: (1+66*x+655*x^2+1596*x^3+1167*x^4+226*x^5+x^6)/(1-x)^6 = 1+8*x*(9+80*x+202*x^2+144*x^3+29*x^4)/(1-x)^6. - Colin Barker, Apr 13 2012