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A212134 Number of (w,x,y,z) with all terms in {1,...,n} and median<=mean.

%I A212134 #18 Aug 08 2025 15:13:03
%S A212134 0,1,12,57,172,405,816,1477,2472,3897,5860,8481,11892,16237,21672,
%T A212134 28365,36496,46257,57852,71497,87420,105861,127072,151317,178872,
%U A212134 210025,245076,284337,328132,376797,430680,490141,555552,627297,705772,791385,884556,985717
%N A212134 Number of (w,x,y,z) with all terms in {1,...,n} and median<=mean.
%C A212134 Also, the number of (w,x,y,z) with all terms in {1,...,n} and median>=mean.
%C A212134 For a guide to related sequences, see A211795.
%H A212134 Colin Barker, <a href="/A212134/b212134.txt">Table of n, a(n) for n = 0..1000</a>
%H A212134 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A212134 a(n)+ A212135(n) = n^4.
%F A212134 a(n) = n*(n^3 + 2*n^2 - 3*n + 2)/2.
%F A212134 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F A212134 G.f.: x*(1 + 7*x + 7*x^2 - 3*x^3) /(1 - x)^5. - _Colin Barker_, Dec 02 2017
%F A212134 E.g.f.: exp(x)*x*(2 + 10*x + 8*x^2 + x^3)/2. - _Stefano Spezia_, Aug 08 2025
%t A212134 t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[Apply[Plus, Rest[Most[Sort[{w, x, y, z}]]]]/2 <= (w + x + y + z)/4, s = s + 1], {w, 1, #}, {x, 1, #}, {y, 1, #},
%t A212134 {z, 1, #}] &[n]; s)]];
%t A212134 Flatten[Map[{t[#]} &, Range[0, 50]]]  (* A212134 *)
%t A212134 (* _Peter J. C. Moses_, May 01 2012 *)
%o A212134 (PARI) concat(0, Vec(x*(1 + 7*x + 7*x^2 - 3*x^3) /(1 - x)^5 + O(x^40))) \\ _Colin Barker_, Dec 02 2017
%Y A212134 Cf. A211795, A212133.
%K A212134 nonn,easy
%O A212134 0,3
%A A212134 _Clark Kimberling_, May 04 2012