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%I #32 Sep 05 2023 13:52:26
%S 1,0,512,0,44032,0,1549824,0,30349312,524288,391784960,26738688,
%T 3702064128,508035072,27315850752,5588385792,164415418368,42844291072,
%U 834237682176,253170810880,3662806408192,1226904698880,14215186707968,5082890895360,49621579941888
%N Coordination sequence for diamond structure D^+_16. (Edges defined by l_1 norm = 1.)
%H Vincenzo Librandi, <a href="/A035884/b035884.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="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_32">Index entries for linear recurrences with constant coefficients</a>, signature (0, 16, 0, -120, 0, 560, 0, -1820, 0, 4368, 0, -8008, 0, 11440, 0, -12870, 0, 11440, 0, -8008, 0, 4368, 0, -1820, 0, 560, 0, -120, 0, 16, 0, -1).
%F G.f.: (x^32 +496*x^30 +35960*x^28 +906192*x^26 +10551068*x^24 +524288*x^23 +68444400*x^22 +18350080*x^21 +285430600*x^20 +143130624*x^19 +733841744*x^18 +374865920*x^17 +1022804550*x^16 +374865920*x^15 +733841744*x^14 +143130624*x^13 +285430600*x^12 +18350080*x^11 +68444400*x^10 +524288*x^9 +10551068*x^8 +906192*x^6 +35960*x^4 +496*x^2 +1) / ((x-1)^16*(x+1)^16). - _Colin Barker_, Feb 26 2013
%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=16.
%t CoefficientList[Series[(x^32 + 496 x^30 + 35960 x^28 + 906192 x^26 + 10551068 x^24 + 524288 x^23 + 68444400 x^22 + 18350080 x^21 + 285430600 x^20 + 143130624 x^19 + 733841744 x^18 + 374865920 x^17 + 1022804550 x^16 + 374865920 x^15 + 733841744 x^14 + 143130624 x^13 + 285430600 x^12 + 18350080 x^11 + 68444400 x^10 + 524288 x^9 + 10551068 x^8 + 906192 x^6 + 35960 x^4 + 496 x^2 + 1)/((x - 1)^16 (x + 1)^16), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 21 2013 *)
%K nonn,easy
%O 0,3
%A Joan Serra-Sagrista (jserra(AT)ccd.uab.es)
%E Recomputed by _N. J. A. Sloane_, Nov 27 1998
%E More terms from _Colin Barker_, Feb 26 2013
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