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A211612 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+x+y>=0.

%I A211612 #14 Dec 05 2017 05:45:45
%S A211612 0,4,35,117,274,530,909,1435,2132,3024,4135,5489,7110,9022,11249,
%T A211612 13815,16744,20060,23787,27949,32570,37674,43285,49427,56124,63400,
%U A211612 71279,79785,88942,98774,109305,120559,132560,145332,158899,173285,188514,204610,221597
%N A211612 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+x+y>=0.
%C A211612 For a guide to related sequences, see A211422.
%H A211612 Colin Barker, <a href="/A211612/b211612.txt">Table of n, a(n) for n = 0..1000</a>
%H A211612 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A211612 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F A211612 From _Colin Barker_, Dec 04 2017: (Start)
%F A211612 G.f.: x*(4 + 19*x + x^2) / (1 - x)^4.
%F A211612 a(n) = (n*(-3 + 3*n + 8*n^2))/2.
%F A211612 (End)
%t A211612 t = Compile[{{u, _Integer}}, Module[{s = 0}, (Do[If[w + x + y >= 0, s = s + 1], {w, #}, {x, #}, {y, #}] &[Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
%t A211612 Map[t[#] &, Range[0, 60]]  (* A211612 *)
%t A211612 FindLinearRecurrence[%]
%t A211612 (* _Peter J. C. Moses_, Apr 13 2012 *)
%t A211612 LinearRecurrence[{4, -6, 4, -1},{0, 4, 35, 117},36] (* _Ray Chandler_, Aug 02 2015 *)
%o A211612 (PARI) concat(0, Vec(x*(4 + 19*x + x^2) / (1 - x)^4 + O(x^40))) \\ _Colin Barker_, Dec 04 2017
%Y A211612 Cf. A211422.
%K A211612 nonn,easy
%O A211612 0,2
%A A211612 _Clark Kimberling_, Apr 16 2012