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A035879 Coordination sequence for diamond structure D^+_6. (Edges defined by l_1 norm = 1.)

%I A035879 #27 Sep 05 2023 13:01:57
%S A035879 1,0,72,32,1104,672,7128,4032,28320,14784,84072,41184,206064,96096,
%T A035879 441336,198016,855360,372096,1535112,651168,2592144,1076768,4165656,
%U A035879 1700160,6425568,2583360,9575592,3800160,13856304,5437152,19548216,7594752,26974848
%N A035879 Coordination sequence for diamond structure D^+_6. (Edges defined by l_1 norm = 1.)
%H A035879 Vincenzo Librandi, <a href="/A035879/b035879.txt">Table of n, a(n) for n = 0..1000</a>
%H A035879 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 A035879 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 A035879 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-15,0,20,0,-15,0,6,0,-1).
%F A035879 G.f.: (x^12 + 66*x^10 + 32*x^9 + 687*x^8 + 480*x^7 + 1564*x^6 + 480*x^5 + 687*x^4 + 32*x^3 + 66*x^2 + 1) / ((x-1)^6*(x+1)^6). [_Colin Barker_, Nov 20 2012]
%F A035879 a(n) = 2*n*( (2*n^4+20*n^2+23)*(-1)^n + (4*n^4+10*n^2+31) )/15 for n>0, a(0)=1. [_Bruno Berselli_, Oct 21 2013]
%p A035879 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=6.
%t A035879 f[m_, n_] := 2^(n - 1)*Binomial[(n + 2*m)/2 - 1, n - 1] + If[EvenQ[m], 2*n*Hypergeometric2F1[1 - m, 1 - n, 2, 2], 0]; f[0, _] = 1; Table[f[m, 6], {m, 0, 27}] (* _Jean-François Alcover_, Apr 18 2013, after Maple *)
%t A035879 CoefficientList[Series[(x^12 + 66 x^10 + 32 x^9 + 687 x^8 + 480 x^7 + 1564 x^6 + 480 x^5 + 687 x^4 + 32 x^3 + 66 x^2 + 1)/((x - 1)^6 (x + 1)^6), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 21 2013 *)
%o A035879 (Magma) [1] cat [2*n*((2*n^4+20*n^2+23)*(-1)^n+(4*n^4+10*n^2+31))/15: n in [1..20]]; // _Bruno Berselli_, Oct 21 2013
%K A035879 nonn,easy
%O A035879 0,3
%A A035879 Joan Serra-Sagrista (jserra(AT)ccd.uab.es)
%E A035879 Recomputed by _N. J. A. Sloane_, Nov 27 1998
%E A035879 More terms from _Vincenzo Librandi_, Oct 21 2013