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A212089 Number of (w,x,y,z) with all terms in {1,...,n} and w>=average{x,y,z}.

%I A212089 #20 Dec 04 2016 19:46:29
%S A212089 0,1,9,45,139,333,684,1258,2133,3402,5167,7542,10656,14647,19665,
%T A212089 25875,33451,42579,53460,66304,81333,98784,118903,141948,168192,
%U A212089 197917,231417,269001,310987,357705,409500,466726,529749,598950,674719
%N A212089 Number of (w,x,y,z) with all terms in {1,...,n} and w>=average{x,y,z}.
%C A212089 Also, number of (w,x,y,z) with all terms in {1,...,n} and w<=average{x,y,z}.
%C A212089 a(n)+A212088(n)=n^4.
%C A212089 For a guide to related sequences, see A211795.
%H A212089 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (4, -6, 5, -5, 6, -4, 1).
%F A212089 a(n) = 4*a(n-1)-6*a(n-2)+5*a(n-3)-5*a(n-4)+6*a(n-5)-4*a(n-6)+a(n-7).
%F A212089 G.f.: x*(1+7*x^4+8*x^3+15*x^2+5*x) / ((x^2+x+1)*(-x+1)^5). - _Alois P. Heinz_, May 18 2012
%t A212089 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212089 (Do[If[3 w >= x + y + z, s = s + 1],
%t A212089 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212089 Map[t[#] &, Range[0, 50]] (* A212088 *)
%t A212089 FindLinearRecurrence[%]
%t A212089 (* _Peter J. C. Moses_, Apr 13 2012 *)
%t A212089 LinearRecurrence[{4, -6, 5, -5, 6, -4, 1},{0, 1, 9, 45, 139, 333, 684},35] (* _Ray Chandler_, Aug 02 2015 *)
%Y A212089 Cf. A211795, A212069, A212088.
%K A212089 nonn
%O A212089 0,3
%A A212089 _Clark Kimberling_, May 01 2012