%I #27 Sep 05 2023 13:51:56
%S 1,0,392,0,25872,0,703640,8192,10861088,860160,113156008,19496960,
%T 873656112,222265344,5301934776,1666990080,26420376640,9372188672,
%U 111885458888,42600857600,413780326736,164317593600,1365594249432,555941191680,4091822419552
%N Coordination sequence for diamond structure D^+_14. (Edges defined by l_1 norm = 1.)
%H Vincenzo Librandi, <a href="/A035883/b035883.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_28">Index entries for linear recurrences with constant coefficients</a>, signature (0, 14, 0, -91, 0, 364, 0, -1001, 0, 2002, 0, -3003, 0, 3432, 0, -3003, 0, 2002, 0, -1001, 0, 364, 0, -91, 0, 14, 0, -1).
%F G.f.: (x^28 +378*x^26 +20475*x^24 +376740*x^22 +8192*x^21 +3222793*x^20 +745472*x^19 +16104998*x^18 +8200192*x^17 +46822139*x^16 +24600576*x^15 +68231544*x^14 +24600576*x^13 +46822139*x^12 +8200192*x^11 +16104998*x^10 +745472*x^9 +3222793*x^8 +8192*x^7 +376740*x^6 +20475*x^4 +378*x^2 +1) / ((x -1)^14*(x +1)^14). [_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=14.
%t CoefficientList[Series[(x^28 + 378 x^26 + 20475 x^24 + 376740 x^22 + 8192 x^21 + 3222793 x^20 + 745472 x^19 + 16104998 x^18 + 8200192 x^17 + 46822139 x^16 + 24600576 x^15 + 68231544 x^14 + 24600576 x^13 + 46822139 x^12 + 8200192 x^11 + 16104998 x^10 + 745472 x^9 + 3222793 x^8 + 8192 x^7 + 376740 x^6 + 20475 x^4 + 378 x^2 + 1)/((x - 1)^14 (x + 1)^14), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 21 2013 *)
%o (PARI) {A035883_vec(N,c=[1, 0, 378, 0, 20475, 0, 376740, 8192, 3222793, 745472, 16104998, 8200192, 46822139, 24600576, 68231544])= Vec(Pol(concat(c,vecextract(c,"-2..1")))/(x^2-1)^14+O(x^N))} \\ _M. F. Hasler_, 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