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

0,2

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).

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

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 *)

Prog

(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
(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

Crossrefs

Row 10 of A103881.

Sequence in context: A185536 A201833 A035836 * A201213 A250528 A251029

Adjacent sequences: A008392 A008393 A008394 * A008396 A008397 A008398

Keyword

nonn,easy

Author

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

Status

approved