%I #29 May 30 2023 14:11:38

%S 1,111,3191,43561,365751,2181257,10106977,38619087,126825227,

%T 368750757,970336269,2349638259,5303497629,11272376259,22745345019,

%U 43859522037,81262792557,145325576067,251806927307

%N Crystal ball sequence for A_10 lattice.

%C Comment from _T. D. Noe_, Apr 29 2007: For the formula to produce this sequence, the five minus signs should be pluses and a 1 should be added: 46189/907200*n^10 + 46189/181440*n^9 + 89947/60480*n^8 + 493207/43200*n^6 + 4812379/181440*n^4 + 43461/2800*n^2 + 26741/6048*n^7 + 171457/8640*n^5 + 111683/4536*n^3 + 7381/1260*n + 1.

%C Comment from _N. J. A. Sloane_, May 04 2007: In that case I need to recheck both the formula and the values to see which is correct.

%H T. D. Noe, <a href="/A008396/b008396.txt">Table of n, a(n) for n = 0..1000</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/Cor#crystal_ball">Index entries for crystal ball sequences</a>

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

%F a(n) = 1 + 11*n*(1+n)*(483120 + 797004*n + 1233596*n^2 + 953849*n^3 + 682786*n^4 + 258791*n^5 + 105859*n^6 + 16796*n^7 + 4199*n^8)/907200.

%F 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)^11. - _Colin Barker_, May 28 2012

%p seq(1 + 11*n*(1+n)*(483120 + 797004*n + 1233596*n^2 + 953849*n^3 + 682786*n^4 + 258791*n^5 + 105859*n^6 + 16796*n^7 + 4199*n^8)/907200, n=0..40);

%t CoefficientList[Series[(x^10+100x^9+2025x^8+14400x^7+44100x^6+63504x^5+44100x^4+14400x^3+2025x^2+100x+1)/(1-x)^11,{x,0,40}],x] (* or *)

%t LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,111,3191,43561, 365751,2181257,10106977,38619087,126825227,368750757,970336269},40] (* _Harvey P. Dale_, Mar 15 2023 *)

%o (Magma) [1 +11*n*(1+n)*(483120 +797004*n +1233596*n^2 +953849*n^3 +682786*n^4 +258791*n^5 +105859*n^6 +16796*n^7 +4199*n^8)/907200: n in [0..40]]; // _G. C. Greubel_, May 29 2023

%o (SageMath)

%o def A008396(n): return 1 +11*n*(1+n)*(483120 +797004*n +1233596*n^2 +953849*n^3 +682786*n^4 +258791*n^5 +105859*n^6 +16796*n^7 +4199*n^8)/907200

%o [A008396(n) for n in range(41)] # _G. C. Greubel_, May 29 2023

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_ and _J. H. Conway_