A212501 - OEIS

A212501

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

0, 0, 2, 13, 45, 115, 245, 462, 798, 1290, 1980, 2915, 4147, 5733, 7735, 10220, 13260, 16932, 21318, 26505, 32585, 39655, 47817, 57178, 67850, 79950, 93600, 108927, 126063, 145145, 166315, 189720, 215512, 243848, 274890, 308805

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OFFSET

0,3

COMMENTS

For a guide to related sequences, see A211795.

Partial sums of A033994. - J. M. Bergot, Jun 14 2013

LINKS

Table of n, a(n) for n=0..35.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

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

From Bruno Berselli, May 31 2012: (Start)

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

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

Sum_{n>=2} 1/a(n) = 302/21 + 50*Pi*tan(Pi/10)/7 - 125*log(5)/7 + 50*sqrt(5)*log(phi)/7, where phi is the golden ratio (A001622). - Amiram Eldar, Jun 20 2026

Sum_{n>=2} (-1)^n/a(n) = 4*(8*log(2) - 25*sqrt(5)*log(phi) - 115/6 + 5^(7/4)*Pi/sqrt(phi))/7. - Vaclav Kotesovec, Jun 20 2026

EXAMPLE

a(8) = 798 which results from the following: 1*(8+9+10+11+12+13+14) + 2*(8+9+10+11+12+13) + 3*(8+9+10+11+12) + 4*(8+9+10+11) + 5*(8+9+10) + 6*(8+9) + 7*(8) = 798 = 77+126+150+152+135+102+56. - J. M. Bergot, Aug 23 2022

MAPLE

A212501:=n->(n-1)*n*(n+1)*(5*n-2)/24: seq(A212501(n), n=0..60); # Wesley Ivan Hurt, Oct 07 2017

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]]

(* Alternative: *)

LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 2, 13, 45}, 50] (* Harvey P. Dale, May 01 2023 *)

PROG

(PARI) a(n)=n*(n-1)*(n+1)*(5*n-2)/24 \\ Charles R Greathouse IV, Jun 14 2013

CROSSREFS