%I #18 Sep 03 2023 15:44:27

%S 1,0,128,0,2944,1024,31616,15360,199424,101376,877696,439296,3011200,

%T 1464320,8630144,4073472,21607936,9922560,48713856,21829632,101009792,

%U 44301312,195640192,84198400,358064384

%N Number of points of L1 norm 2n in root system version of E_8 lattice.

%C Also, coordination sequence for diamond structure D^+_8. (Edges defined by l_1 norm = 1.) - J. Serra-Sagrista (jserra(AT)ccd.uab.es). Confirmed by _N. J. A. Sloane_ Nov 27 1998.

%H Ray Chandler, <a href="/A010369/b010369.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 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 P. Solé, <a href="https://doi.org/10.1016/0012-365X(94)00142-6">Counting lattice points in pyramids</a>, Discr. Math. 139 (1995), 381-392.

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0, 8, 0, -28, 0, 56, 0, -70, 0, 56, 0, -28, 0, 8, 0, -1).

%F G.f.: (1/2)*((1+z^2)^8+256*z^8)/(1-z^2)^8 + (1/2)*(1-z^2)^8/(1+z^2)^8.

%p 1/2*((1+z^2)^8+256*z^8)/(1-z^2)^8+1/2*(1-z^2)^8/(1+z^2)^8

%p f := proc(m) local k,t1; t1 := 2^(n-1)*binomial((n+2*m)/2-1,n-1); if m mod 2 = 0 then t1 := t1+add(2^k*binomial(n,k)*binomial(m-1,k-1),k=0..n); fi; t1; end; where n=8.

%Y Cf. A010368.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, _Simon Plouffe_