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A212415 Number of (w,x,y,z) with all terms in {1,...,n} and w=y<=z.

%I A212415 #21 Jul 31 2023 11:03:30
%S A212415 0,0,3,17,55,135,280,518,882,1410,2145,3135,4433,6097,8190,10780,
%T A212415 13940,17748,22287,27645,33915,41195,49588,59202,70150,82550,96525,
%U A212415 112203,129717,149205,170810,194680,220968,249832,281435,315945
%N A212415 Number of (w,x,y,z) with all terms in {1,...,n} and w<x>=y<=z.
%C A212415 For a guide to related sequences, see A211795.
%C A212415 Partial sums of A162147. - _J. M. Bergot_, Jun 21 2013
%H A212415 G. C. Greubel, <a href="/A212415/b212415.txt">Table of n, a(n) for n = 0..1000</a>
%H A212415 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 A212415 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A212415 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F A212415 From _Bruno Berselli_, May 30 2012: (Start)
%F A212415 G.f.: x^2*(3+2*x)/(1-x)^5.
%F A212415 a(n) = (n-1)*n*(n+1)*(5*n+2)/24. (End)
%F A212415 E.g.f.: x^2*(36 + 32*x + 5*x^2)*exp(x)/24. - _G. C. Greubel_, Jul 11 2019
%e A212415 a(3) counts these (w,x,y,z): (1,2,2,2), (1,2,2,3), (1,3,3,3).
%t A212415 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212415 (Do[If[w < x >= y <= z, s = s + 1],
%t A212415 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212415 Map[t[#] &, Range[0, 40]]   (* A212415 *)
%t A212415 Table[n*(5*n+2)*(n^2-1)/4!, {n,0,40}] (* _G. C. Greubel_, Jul 11 2019 *)
%o A212415 (PARI) vector(40, n, n--; n*(5*n+2)*(n^2-1)/4!) \\ _G. C. Greubel_, Jul 11 2019
%o A212415 (Magma) [n*(5*n+2)*(n^2-1)/24: n in [0..40]]; // _G. C. Greubel_, Jul 11 2019
%o A212415 (Sage) [n*(5*n+2)*(n^2-1)/24 for n in (0..40)] # _G. C. Greubel_, Jul 11 2019
%o A212415 (GAP) List([0..40], n-> n*(5*n+2)*(n^2-1)/24); # _G. C. Greubel_, Jul 11 2019
%Y A212415 Cf. A211795.
%K A212415 nonn,easy
%O A212415 0,3
%A A212415 _Clark Kimberling_, May 19 2012