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%I #35 Sep 08 2022 08:44:52
%S 0,2,48,578,4672,28610,142000,596610,2187520,7159170,21278640,
%T 58227906,148321344,354870594,803467056,1732242690,3575055360,
%U 7094825730,13591279920,25216532290,45443741760,79749022402
%N Number of points of L1 norm 12 in cubic lattice Z^n.
%H Vincenzo Librandi, <a href="/A035606/b035606.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. - _N. J. A. Sloane_, Feb 13 2013
%H Joan Serra-Sagrista, <a href="https://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_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
%F From _Colin Barker_, Apr 15 2012: (Start)
%F a(n) = 2*n^2*(146430 + 239327*n^2 + 74800*n^4 + 6996*n^6 + 220*n^8 + 2*n^10)/467775.
%F G.f.: 2*x*(1+x)^11/(1-x)^13. (End)
%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)^11/(1-x)^13,{x,0,30}],x](* _Vincenzo Librandi_, Apr 24 2012 *)
%t LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{0,2,48,578,4672,28610,142000,596610,2187520,7159170,21278640,58227906,148321344},40] (* _Harvey P. Dale_, May 29 2021 *)
%o (Magma) I:=[0, 2, 48, 578, 4672, 28610, 142000, 596610, 2187520, 7159170, 21278640, 58227906, 148321344]; [n le 13 select I[n] else 13*Self(n-1)-78*Self(n-2)+286*Self(n-3)-715*Self(n-4)+1287*Self(n-5)-1716*Self(n-6)+1716*Self(n-7)-1287*Self(n-8)+715*Self(n-9)-286*Self(n-10)+78*Self(n-11)-13*Self(n-12)+Self(n-13): n in [1..30]]; // _Vincenzo Librandi_, Apr 24 2012
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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