A212415 - OEIS

%I #21 Jul 31 2023 11:03:30

%S 0,0,3,17,55,135,280,518,882,1410,2145,3135,4433,6097,8190,10780,

%T 13940,17748,22287,27645,33915,41195,49588,59202,70150,82550,96525,

%U 112203,129717,149205,170810,194680,220968,249832,281435,315945

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

%C For a guide to related sequences, see A211795.

%C Partial sums of A162147. - _J. M. Bergot_, Jun 21 2013

%H G. C. Greubel, <a href="/A212415/b212415.txt">Table of n, a(n) for n = 0..1000</a>

%H Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, <a href="https://arxiv.org/abs/2307.13749">Semi-simplicial combinatorics of cyclinders and subdivisions</a>, arXiv:2307.13749 [math.CO], 2023. See p. 25.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

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

%F From _Bruno Berselli_, May 30 2012: (Start)

%F G.f.: x^2*(3+2*x)/(1-x)^5.

%F a(n) = (n-1)*n*(n+1)*(5*n+2)/24. (End)

%F E.g.f.: x^2*(36 + 32*x + 5*x^2)*exp(x)/24. - _G. C. Greubel_, Jul 11 2019

%e a(3) counts these (w,x,y,z): (1,2,2,2), (1,2,2,3), (1,3,3,3).

%t t = Compile[{{n, _Integer}}, Module[{s = 0},

%t (Do[If[w < x >= y <= z, s = s + 1],

%t {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];

%t Map[t[#] &, Range[0, 40]]   (* A212415 *)

%t Table[n*(5*n+2)*(n^2-1)/4!, {n,0,40}] (* _G. C. Greubel_, Jul 11 2019 *)

%o (PARI) vector(40, n, n--; n*(5*n+2)*(n^2-1)/4!) \\ _G. C. Greubel_, Jul 11 2019

%o (Magma) [n*(5*n+2)*(n^2-1)/24: n in [0..40]]; // _G. C. Greubel_, Jul 11 2019

%o (Sage) [n*(5*n+2)*(n^2-1)/24 for n in (0..40)] # _G. C. Greubel_, Jul 11 2019

%o (GAP) List([0..40], n-> n*(5*n+2)*(n^2-1)/24); # _G. C. Greubel_, Jul 11 2019

%Y Cf. A211795.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, May 19 2012