%I #25 Sep 08 2022 08:44:36
%S 1,57,995,8683,48429,198677,653791,1825719,4494329,10016209,20601723,
%T 39670115,72292453,125732205,210093239,339085039,530914929,809317097,
%U 1204728211,1755619419,2509994525,3527064133,4879105551,6653518247,8955084649,11908446081,15660803627,20384853715
%N Crystal ball sequence for lattice {E_7}*.
%H T. D. Noe, <a href="/A008922/b008922.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Cor#crystal_ball">Index entries for crystal ball sequences</a>
%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 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F G.f.: (1+x)*(1 +48*x +519*x^2 +1744*x^3 +1959*x^4 +624*x^5 +x^6)/(1-x)^8.
%F From _G. C. Greubel_, Sep 13 2019: (Start)
%F a(n) = (35 +37*n +679*n^2 +3928*n^3 +280*n^4 +518*n^5 -14*n^6 +68*n^7)/35.
%F E.g.f.: (35 +1960*x +154358x^2 +34230*x^3 +28350*x^4 +9828*x^5 +1414*x^6 +68*x^7)*exp(x)/35. (End)
%p seq((35 +37*n +679*n^2 +392*n^3 +280*n^4 +518*n^5 -14*n^6 +68*n^7)/35, n=0..30); # _G. C. Greubel_, Sep 13 2019
%t CoefficientList[Series[(1+x)(x^6+624x^5+1959x^4+1744x^3+519x^2+48x+1)/ (1-x)^8,{x,0,30}],x] (* _Harvey P. Dale_, May 14 2019 *)
%o (PARI) a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; -1,8,-28,56,-70,56,-28,8]^n*[1;57;995;8683;48429;198677;653791;1825719])[1,1] \\ _Charles R Greathouse IV_, Feb 10 2017
%o (PARI) vector(30, n, m=n-1; (35 +37*m +679*m^2 +392*m^3 +280*m^4 +518*m^5 -14*m^6 +68*m^7)/35) \\ _G. C. Greubel_, Sep 13 2019
%o (Magma) [(35 +37*n +679*n^2 +392*n^3 +280*n^4 +518*n^5 -14*n^6 +68*n^7)/35: n in [0..30]]; // _G. C. Greubel_, Sep 13 2019
%o (Sage) [(35 +37*n +679*n^2 +392*n^3 +280*n^4 +518*n^5 -14*n^6 +68*n^7)/35 for n in (0..30)] # _G. C. Greubel_, Sep 13 2019
%o (GAP) List([0..30], n-> (35 +37*n +679*n^2 +392*n^3 +280*n^4 +518*n^5 -14*n^6 +68*n^7)/35); # _G. C. Greubel_, Sep 13 2019
%K nonn,nice,easy
%O 0,2
%A _N. J. A. Sloane_