%I #41 Dec 10 2023 16:14:02

%S 1,14,98,462,1596,4410,10374,21658,41272,73206,122570,195734,300468,

%T 446082,643566,905730,1247344,1685278,2238642,2928926,3780140,4818954,

%U 6074838,7580202,9370536,11484550,13964314,16855398,20207012,24072146,28507710,33574674,39338208,45867822

%N Coordination sequence for {A_6}* lattice.

%C Equally, coordination sequence for 6-dimensional cyclotomic lattice Z[zeta_14].

%D M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.

%H Vincenzo Librandi, <a href="/A008534/b008534.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Beck and S. Hosten, <a href="http://arxiv.org/abs/math/0508136">Cyclotomic polytopes and growth series of cyclotomic lattices</a>, arXiv:math/0508136 [math.CO], 2005-2006.

%H J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).

%H M. O'Keeffe, <a href="http://dx.doi.org/10.1524/zkri.1995.210.12.905">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908.

%H M. O'Keeffe, <a href="/A008527/a008527.pdf">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F G.f.: (x^6+8*x^5+29*x^4+64*x^3+29*x^2+8*x+1)/(x-1)^6. [Conway-Sloane] - _Colin Barker_, Sep 21 2012

%F a(n) = (7/6)*n*(n^2+2)*(n^2+3) for n>0, a(0)=1. - _Bruno Berselli_, Feb 28 2013

%F E.g.f.: 1 + x*(84 + 210*x + 210*x^2 + 70*x^3 + 7*x^4)*exp(x)/6. - _G. C. Greubel_, Nov 10 2019

%p 1, seq( (7*k^5+35*k^3+42*k)/6, k=1..40);

%t CoefficientList[Series[(x^6 +8x^5 +29x^4 +64x^3 +29x^2 +8x +1)/(x-1)^6, {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 20 2013 *)

%t Table[If[n==0,1, 7*n*(6+5*n^2+n^4)/6], {n,0,40}] (* _G. C. Greubel_, Nov 10 2019 *)

%o (PARI) vector(46, n, if(n==1,1, 7*(n-1)*(6+5*(n-1)^2+(n-1)^4)/6 ) ) \\ _G. C. Greubel_, Nov 10 2019

%o (Magma) [1] cat [7*n*(6+5*n^2+n^4)/6: n in [1..45]]; // _G. C. Greubel_, Nov 10 2019

%o (Sage) [1]+[7*n*(6+5*n^2+n^4)/6 for n in (1..45)]; # _G. C. Greubel_, Nov 10 2019

%o (GAP) Concatenation([1], List([1..45], n-> 7*n*(6+5*n^2+n^4)/6 )); # _G. C. Greubel_, Nov 10 2019

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_