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

0, 0, 3, 17, 55, 135, 280, 518, 882, 1410, 2145, 3135, 4433, 6097, 8190, 10780, 13940, 17748, 22287, 27645, 33915, 41195, 49588, 59202, 70150, 82550, 96525, 112203, 129717, 149205, 170810, 194680, 220968, 249832, 281435, 315945

Offset: 0

Author

Clark Kimberling, May 19 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Partial sums of A162147. - J. M. Bergot, Jun 21 2013

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, Semi-simplicial combinatorics of cyclinders and subdivisions, arXiv:2307.13749 [math.CO], 2023. See p. 25.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A211795.

Programs

GAP

List([0..40], n-> n*(5*n+2)*(n^2-1)/24); # G. C. Greubel, Jul 11 2019

Magma

[n*(5*n+2)*(n^2-1)/24: n in [0..40]]; // G. C. Greubel, Jul 11 2019

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < x >= y <= z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212415 *)
Table[n*(5*n+2)*(n^2-1)/4!, {n,0,40}] (* G. C. Greubel, Jul 11 2019 *)

PARI

vector(40, n, n--; n*(5*n+2)*(n^2-1)/4!) \\ G. C. Greubel, Jul 11 2019

Sage

[n*(5*n+2)*(n^2-1)/24 for n in (0..40)] # G. C. Greubel, Jul 11 2019

Formula

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

From Bruno Berselli, May 30 2012: (Start)

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

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

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