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A212522 Number of (w,x,y,z) with all terms in {1,...,n} and w>=2x and y>3z.

%I A212522 #8 Dec 11 2015 07:57:58
%S A212522 0,0,0,0,4,12,27,60,112,180,300,450,648,924,1274,1680,2240,2880,3645,
%T A212522 4590,5700,6930,8470,10164,12096,14352,16900,19656,22932,26460,30375,
%U A212522 34800,39680,44880,50864,57222,64152,71820,80142,88920,98800,109200,120393
%N A212522 Number of (w,x,y,z) with all terms in {1,...,n} and w>=2x and y>3z.
%C A212522 For a guide to related sequences, see A211795.
%H A212522 Colin Barker, <a href="/A212522/b212522.txt">Table of n, a(n) for n = 0..1000</a>
%H A212522 <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 A212522 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 A212522 G.f.: x^4*(4 +12*x +19*x^2 +28*x^3 +38*x^4 +34*x^5 +31*x^6 +26*x^7 +16*x^8 +6*x^9 +2*x^10) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3). - _Colin Barker_, Dec 11 2015
%t A212522 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212522 (Do[If[w >= 2 x && y > 3 z, s = s + 1],
%t A212522 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212522 Map[t[#] &, Range[0, 50]]   (* A212522 *)
%o A212522 (PARI) concat(vector(4), Vec(x^4*(4 +12*x +19*x^2 +28*x^3 +38*x^4 +34*x^5 +31*x^6 +26*x^7 +16*x^8 +6*x^9 +2*x^10) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3) + O(x^100))) \\ _Colin Barker_, Dec 11 2015
%Y A212522 Cf. A211795, A212508.
%K A212522 nonn,easy
%O A212522 0,5
%A A212522 _Clark Kimberling_, May 20 2012