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

0, 1, 12, 57, 172, 405, 816, 1477, 2472, 3897, 5860, 8481, 11892, 16237, 21672, 28365, 36496, 46257, 57852, 71497, 87420, 105861, 127072, 151317, 178872, 210025, 245076, 284337, 328132, 376797, 430680, 490141, 555552, 627297, 705772, 791385, 884556, 985717

Offset: 0

Author

Clark Kimberling, May 04 2012

Keywords

• nonn
• easy

Comments

Also, the number of (w,x,y,z) with all terms in {1,...,n} and median>=mean.

For a guide to related sequences, see A211795.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A211795, A212133.

Programs

Mathematica

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, #},
{z, 1, #}] &[n]; s)]];
Flatten[Map[{t[#]} &, Range[0, 50]]]  (* A212134 *)
(* Peter J. C. Moses, May 01 2012 *)

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

Formula

a(n)+ A212135(n) = n^4.

a(n) = n*(n^3 + 2*n^2 - 3*n + 2)/2.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

G.f.: x*(1 + 7*x + 7*x^2 - 3*x^3) /(1 - x)^5. - Colin Barker, Dec 02 2017

E.g.f.: exp(x)*x*(2 + 10*x + 8*x^2 + x^3)/2. - Stefano Spezia, Aug 08 2025