cp's OEIS Frontend

A212507 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y<=2z.

%I A212507 #18 Jan 05 2025 05:53:45
%S A212507 0,1,12,56,168,399,810,1480,2496,3965,6000,8736,12312,16891,22638,
%T A212507 29744,38400,48825,61236,75880,93000,112871,135762,161976,191808,
%U A212507 225589,263640,306320,353976,406995,465750,530656,602112,680561,766428
%N A212507 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y<=2z.
%C A212507 For a guide to related sequences, see A211795.
%H A212507 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).
%F A212507 a(n) = 3*a(n-1)-a(n-2)-5*a(n-3)+5*a(n-4)+a(n-5)-3*a(n-6)+a(n-7).
%F A212507 G.f.: x*(1+2*x)*(1+7*x+7*x^2+3*x^3)/((1+x)^2*(1-x)^5). [_Bruno Berselli_, May 31 2012]
%F A212507 a(n) = (2*n*(9*n^3+6*n^2+1)-(2*n-1)*(-1)^n-1)/32. [_Bruno Berselli_, May 31 2012]
%F A212507 a(n) = A006578(n) * A077043(n). - _Alois P. Heinz_, May 31 2012
%t A212507 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212507 (Do[If[w < 2 x && y <= 2 z, s = s + 1],
%t A212507 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212507 Map[t[#] &, Range[0, 40]]   (* A212507 *)
%t A212507 CoefficientList[Series[x (1 + 2 x) (1 + 7 x + 7 x^2 + 3 x^3)/((1 + x)^2 (1 - x)^5), {x, 0, 34}], x] (* _Bruno Berselli_, May 31 2012 *)
%o A212507 (Magma) [(2*n*(9*n^3+6*n^2+1)-(2*n-1)*(-1)^n-1)/32: n in [0..34]]; // _Bruno Berselli_, May 31 2012
%Y A212507 Cf. A211795.
%K A212507 nonn,easy
%O A212507 0,3
%A A212507 _Clark Kimberling_, May 19 2012