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A212514 Number of (w,x,y,z) with all terms in {1,...,n} and w<=2x and y>3z.

%I A212514 #10 Dec 24 2020 13:59:39
%S A212514 0,0,0,0,14,42,90,200,364,585,960,1440,2052,2926,4004,5280,7000,9000,
%T A212514 11340,14280,17670,21483,26180,31416,37296,44252,52000,60480,70434,
%U A212514 81270,93150,106720,121520,137445,155584,175032,196020,219450,244644,271440,301340
%N A212514 Number of (w,x,y,z) with all terms in {1,...,n} and w<=2x and y>3z.
%C A212514 For a guide to related sequences, see A211795.
%H A212514 Colin Barker, <a href="/A212514/b212514.txt">Table of n, a(n) for n = 0..1000</a>
%H A212514 <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 A212514 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 A212514 G.f.: x^4*(14 +42*x +62*x^2 +88*x^3 +114*x^4 +103*x^5 +90*x^6 +74*x^7 +42*x^8 +15*x^9 +4*x^10) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3). - _Colin Barker_, Dec 18 2015
%t A212514 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212514 (Do[If[w <= 2 x && y > 3 z, s = s + 1],
%t A212514 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212514 Map[t[#] &, Range[0, 50]]   (* A212514 *)
%t A212514 LinearRecurrence[{0,2,2,-1,-4,0,2,0,-2,0,4,1,-2,-2,0,1},{0,0,0,0,14,42,90,200,364,585,960,1440,2052,2926,4004,5280},50] (* _Harvey P. Dale_, Dec 24 2020 *)
%o A212514 (PARI) concat(vector(4), Vec(x^4*(14 +42*x +62*x^2 +88*x^3 +114*x^4 +103*x^5 +90*x^6 +74*x^7 +42*x^8 +15*x^9 +4*x^10) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3) + O(x^60))) \\ _Colin Barker_, Dec 18 2015
%Y A212514 Cf. A211795, A212508.
%K A212514 nonn,easy
%O A212514 0,5
%A A212514 _Clark Kimberling_, May 20 2012