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A008535 Coordination sequence for {A_7}* lattice.

%I A008535 #27 Jun 04 2023 12:08:24
%S A008535 1,16,128,688,2746,8752,23536,55568,118498,232976,428752,747056,
%T A008535 1243258,1989808,3079456,4628752,6781826,9714448,13638368,18805936,
%U A008535 25515002,34114096,45007888,58662928,75613666,96468752,121917616,152737328,189799738,234078896
%N A008535 Coordination sequence for {A_7}* lattice.
%H A008535 Colin Barker, <a href="/A008535/b008535.txt">Table of n, a(n) for n = 0..1000</a>
%H A008535 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As7.html">Home page for this lattice</a>
%H A008535 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 A008535 M. O'Keeffe, <a href="/A008527/a008527.pdf">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
%H A008535 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A008535 G.f.: (1+x)*(1+8*x+29*x^2+64*x^3+29*x^4+8*x^5+x^6)/(1-x)^7. - _Colin Barker_, Mar 03 2015
%F A008535 E.g.f.: -1 + (36 + 252*x + 882*x^2 + 1050*x^3 + 525*x^4 + 105*x^5 + 7*x^6)*exp(x)/18. - _G. C. Greubel_, Nov 10 2019
%p A008535 1, seq( (7*k^6+70*k^4+175*k^2+36)/18, k=1..40);
%t A008535 Table[If[n==0,1,(36+175*n^2+70*n^4+7*n^6)/18], {n,0,40}] (* _G. C. Greubel_, Nov 10 2019 *)
%t A008535 LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,16,128,688,2746,8752,23536,55568},40] (* _Harvey P. Dale_, Jun 04 2023 *)
%o A008535 (PARI) Vec(-(x+1)*(x^6+8*x^5+29*x^4+64*x^3+29*x^2+8*x+1) / (x-1)^7 + O(x^40)) \\ _Colin Barker_, Mar 03 2015
%o A008535 (PARI) vector(46, n, if(n==1,1, (36+175*(n-1)^2+70*(n-1)^4+7*(n-1)^6)/18 ) ) \\ _G. C. Greubel_, Nov 10 2019
%o A008535 (Magma) [1] cat [(36+175*n^2+70*n^4+7*n^6)/18: n in [1..45]]; // _G. C. Greubel_, Nov 10 2019
%o A008535 (Sage) [1]+[(36+175*n^2+70*n^4+7*n^6)/18 for n in (1..45)]; # _G. C. Greubel_, Nov 10 2019
%o A008535 (GAP) Concatenation([1], List([1..45], n-> (36+175*n^2+70*n^4+7*n^6)/18 )); # _G. C. Greubel_, Nov 10 2019
%K A008535 nonn,easy
%O A008535 0,2
%A A008535 _N. J. A. Sloane_