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

Offset: 0

Author

Clark Kimberling, May 19 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

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

Examples

			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
		

Links

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

Crossrefs

Cf. A211795.

Programs

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]]   (* A212501 *)
LinearRecurrence[{5,-10,10,-5,1},{0,0,2,13,45},50] (* Harvey P. Dale, May 01 2023 *)

PARI

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

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.

G.f.: x^2*(2+3*x)/(1-x)^5. - Bruno Berselli, May 31 2012

a(n) = (n-1)*n*(n+1)*(5*n-2)/24. - Bruno Berselli, May 31 2012