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A212509 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y<=3z.

%I A212509 #13 Dec 18 2015 11:33:55
%S A212509 0,1,12,63,180,437,891,1628,2736,4392,6600,9646,13608,18669,24990,
%T A212509 32955,42432,54033,67797,84010,102900,125118,150282,179444,212544,
%U A212509 249977,292032,339687,392196,451165,516375,588336,667392,754908,849660,953922,1067256
%N A212509 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y<=3z.
%C A212509 For a guide to related sequences, see A211795.
%H A212509 Colin Barker, <a href="/A212509/b212509.txt">Table of n, a(n) for n = 0..1000</a>
%H A212509 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,2,-1,-4,0,2,0,-2,0,4,1,-2,-2,0,1).
%F A212509 a(n) = 2*a(n-2)+2*a(n-3)-a(n-4)-4*a(n-5)+2*a(n-7)-2*a(n-9)+4*a(n-11)+a(n-12)-2*a(n-13)-2*a(n-14)+a(n-16).
%F A212509 G.f.: x*(1 +12*x +61*x^2 +154*x^3 +288*x^4 +421*x^5 +505*x^6 +510*x^7 +487*x^8 +387*x^9 +246*x^10 +120*x^11 +42*x^12 +6*x^13) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3). - _Colin Barker_, Dec 17 2015
%t A212509 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212509 (Do[If[w < 2 x && y <= 3 z, s = s + 1],
%t A212509 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212509 Map[t[#] &, Range[0, 50]]   (* A212509 *)
%o A212509 (PARI) concat(0, Vec(x*(1 +12*x +61*x^2 +154*x^3 +288*x^4 +421*x^5 +505*x^6 +510*x^7 +487*x^8 +387*x^9 +246*x^10 +120*x^11 +42*x^12 +6*x^13) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3) + O(x^60))) \\ _Colin Barker_, Dec 17 2015
%Y A212509 Cf. A211795, A212508.
%K A212509 nonn,easy
%O A212509 0,3
%A A212509 _Clark Kimberling_, May 20 2012