%I M5419 #39 Dec 10 2023 16:03:59
%S 1,241,9361,131041,996001,5109841,20015281,64495681,179375041,
%T 444798001,1006201681,2111519521,4162485601,7783236241,13909734001,
%U 23903867521,39696408961,63963339121,100340378641,153680892001
%N Crystal ball sequence for E_8 lattice.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A008349/b008349.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 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/Cor#crystal_ball">Index entries for crystal ball sequences</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).
%F G.f.: (1 + 232*x + 7228*x^2 + 55384*x^3 + 133510*x^4 + 107224*x^5 + 24508*x^6 + 232*x^7 + x^8)/(1 - x)^9.
%F a(0)=1, a(1)=241, a(2)=9361, a(3)=131041, a(4)=996001, a(5)=5109841, a(6)=20015281, a(7)=64495681, a(8)=179375041, a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9). - _Harvey P. Dale_, Jun 12 2012
%p 57/7*n^8+108/7*n^7+30*n^6+72*n^5+39*n^4+36*n^3+300/7*n^2-24/7*n+1;
%t CoefficientList[Series[(1+232x+7228x^2+107224x^5+133510x^4+ 55384x^3+ 24508x^6+ 232x^7+ x^8)/(1-x)^9,{x,0,30}],x] (* or *) LinearRecurrence[ {9,-36,84,-126,126,-84,36,-9,1},{1,241,9361,131041,996001,5109841,20015281,64495681,179375041},30] (* _Harvey P. Dale_, Jun 12 2012 *)
%o (Python)
%o A008349_list, m = [], [328320, -1071360, 1347840, -812160, 233280, -25920, 240, 0, 1]
%o for _ in range(10**2):
%o A008349_list.append(m[-1])
%o for i in range(8):
%o m[i+1] += m[i] # _Chai Wah Wu_, Dec 15 2015
%o (Magma) [57/7*n^8 + 108/7*n^7 + 30*n^6 + 72*n^5 + 39*n^4 + 36*n^3 + 300/7*n^2 - 24/7*n + 1: n in [0..40]]; // _Vincenzo Librandi_, Dec 16 2015
%o (PARI) a(n)=(57*n^8 + 108*n^7 + 210*n^6 + 504*n^5 + 273*n^4 + 252*n^3 + 300*n^2 - 24*n + 7)/7 \\ _Charles R Greathouse IV_, Feb 10 2017
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_ and _J. H. Conway_
%E The values given by O'Keeffe are incorrect.