A008395 Coordination sequence for A_10 lattice.

1, 110, 3080, 40370, 322190, 1815506, 7925720, 28512110, 88206140, 241925530, 601585512, 1379301990, 2953859370, 5968878630, 11472968760, 21114177018, 37403270520, 64062783510, 106481351240, 172295622730, 272125000774, 420487598410

Offset: 0

Author

N. J. A. Sloane and J. H. Conway

Keywords

• nonn
• easy

Links

• T. D. Noe, 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, 1997; 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.
• 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 (10,-45,120,-210,252,-210,120,-45,10,-1).

Crossrefs

Row 10 of A103881.

Programs

Magma

[1] cat [11*n*(4199*n^8 +72930*n^6 +327327*n^4 +406120*n^2 +96624)/90720: n in [1..40]]; // G. C. Greubel, May 27 2023

Maple

a:= n-> `if`(n=0, 1, 46189/90720*n^9+26741/3024*n^7+
         171457/4320*n^5+111683/2268*n^3+7381/630*n):
seq(a(n), n=0..25);

Mathematica

a[n_]:= If[n==0, 1, 11n(4199n^8 +72930n^6 +327327n^4 +406120n^2 +96624)/90720];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jan 07 2019 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1}, {1,110,3080, 40370,322190,1815506,7925720,28512110,88206140,241925530,601585512}, 30] (* Harvey P. Dale, Nov 27 2019 *)

SageMath

[11*n*(4199*n^8 +72930*n^6 +327327*n^4 +406120*n^2 +96624)//90720 +int(n==0) for n in range(41)] # G. C. Greubel, May 27 2023

Formula

a(n) = 46189/90720*n^9 +26741/3024*n^7 +171457/4320*n^5 +111683/2268*n^3 +7381/630*n for n >= 1.

Sum_{d=1}^10 C(11, d) C(m/2-1, d-1) C(10-d+m/2, m/2), where norm m is always even. (Serra-Sagrista)

G.f.: (1 +100*x +2025*x^2 +14400*x^3 +44100*x^4 +63504*x^5 +44100*x^6 +14400*x^7 +2025*x^8 +100*x^9 +x^10)/(1-x)^10. - Colin Barker, Sep 26 2012