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A212515 Number of (w,x,y,z) with all terms in {1,...,n} and w<=2x and y>=3z.

%I A212515 #8 Dec 18 2015 07:55:23
%S A212515 0,0,0,8,28,63,150,280,468,780,1200,1728,2508,3458,4620,6160,8000,
%T A212515 10125,12852,15960,19530,23870,28798,34272,40848,48100,56160,65520,
%U A212515 75852,87075,100050,114080,129360,146608,165308,185328,207900,231990,257868,286520
%N A212515 Number of (w,x,y,z) with all terms in {1,...,n} and w<=2x and y>=3z.
%C A212515 For a guide to related sequences, see A211795.
%H A212515 Colin Barker, <a href="/A212515/b212515.txt">Table of n, a(n) for n = 0..1000</a>
%H A212515 <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 A212515 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 A212515 G.f.: x^3*(8 +28*x +47*x^2 +78*x^3 +106*x^4 +102*x^5 +95*x^6 +90*x^7 +56*x^8 +26*x^9 +10*x^10 +2*x^11) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3). - _Colin Barker_, Dec 18 2015
%t A212515 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212515 (Do[If[w <= 2 x && y >= 3 z, s = s + 1],
%t A212515 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212515 Map[t[#] &, Range[0, 50]]   (* A212515 *)
%o A212515 (PARI) concat(vector(3), Vec(x^3*(8 +28*x +47*x^2 +78*x^3 +106*x^4 +102*x^5 +95*x^6 +90*x^7 +56*x^8 +26*x^9 +10*x^10 +2*x^11) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3) + O(x^60))) \\ _Colin Barker_, Dec 18 2015
%Y A212515 Cf. A211795, A212508.
%K A212515 nonn,easy
%O A212515 0,4
%A A212515 _Clark Kimberling_, May 20 2012