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%I #42 Sep 08 2022 08:44:52
%S 0,2,36,326,1992,9290,35436,115598,332688,864146,2060980,4573910,
%T 9545560,18892250,35704060,64797470,113461024,192441122,317222212,
%U 509663334,800061160,1229718378,1854105484,2746713774,4003707568
%N Number of points of L1 norm 9 in cubic lattice Z^n.
%H Vincenzo Librandi, <a href="/A035603/b035603.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. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From N. J. A. Sloane, Feb 13 2013
%H M. Janjic, B. Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5.
%H Joan Serra-Sagrista, <a href="http://dx.doi.org/10.1016/S0020-0190(00)00119-8">Enumeration of lattice points in l_1 norm</a>, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F a(n) = (4*n^9 + 168*n^7 + 1596*n^5 + 3272*n^3 + 630*n)/(5*7*9*9). - _Frank Ellermann_, Mar 16 2002
%F G.f.: 2*x*(1+x)^8/(1-x)^10. - _Colin Barker_, Apr 15 2012
%F a(n) = 2*A099196(n). - _R. J. Mathar_, Dec 10 2013
%p f := proc(d,m) local i; sum( 2^i*binomial(d,i)*binomial(m-1,i-1),i=1..min(d,m)); end; # n=dimension, m=norm
%t CoefficientList[Series[2*x*(1+x)^8/(1-x)^10,{x,0,30}],x] (* _Vincenzo Librandi_, Apr 24 2012 *)
%t LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,2,36,326,1992,9290,35436,115598,332688,864146},30] (* _Harvey P. Dale_, Jan 17 2021 *)
%o (PARI) a(n)=(4*n^9+168*n^7+1596*n^5+3272*n^3+630*n)/2835 \\ _Charles R Greathouse IV_, Dec 07 2011
%o (Magma) [(4*n^9+168*n^7+1596*n^5+3272*n^3+630*n)/2835: n in [0..30]]; // _Vincenzo Librandi_, Apr 24 2012
%Y Cf. A035596-A035607.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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