A028724 a(n) = (1/2)*floor(n/2)*floor((n-1)/2)*floor((n-2)/2).

0, 0, 0, 0, 1, 2, 6, 9, 18, 24, 40, 50, 75, 90, 126, 147, 196, 224, 288, 324, 405, 450, 550, 605, 726, 792, 936, 1014, 1183, 1274, 1470, 1575, 1800, 1920, 2176, 2312, 2601, 2754, 3078, 3249, 3610, 3800, 4200, 4410, 4851, 5082, 5566, 5819, 6348, 6624, 7200

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of symmetric Dyck paths of semilength n and having four peaks. E.g., a(5)=2 because we have UU*DU*DU*DU*DD and U*DUU*DU*DDU*D, where U=(1,1), D=(1,-1) and * indicates peaks. - Emeric Deutsch, Jan 12 2004

Starting with "1" = triangle A171608 * the triangular numbers. - Gary W. Adamson, Dec 12 2009

Integer solutions of (x + y)^3 = (x - y)^2. If x = a(2*n + 2) then y = -a(2*n + 1). y and x may be interchanged. - Thomas Scheuerle, Mar 22 2023

2*a(n+3) interleaves the positive integers of A011379 and A045991. - J.S. Seneschal, Mar 31 2025

Examples

			a(7) = 9 since the 9 tuples [x, y, z, w] in {[4, 3, 2, 2] [4, 3, 3, 2] [4, 3, 3, 3] [4, 3, 4, 2] [4, 3, 4, 3] [5, 2, 2, 2] [5, 2, 3, 2] [5, 2, 4, 2] [5, 2, 5, 2]} are all the solutions of 7 = x + y, x >= max(y, z), min(y, z) >= w >= 2.
		

References

• P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 185, Article 433.

Links

• Iain Fox, Table of n, a(n) for n = 0..10000
• Paul Barry, On sequences with {-1, 0, 1} Hankel transforms, arXiv preprint arXiv:1205.2565 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 18 2012
• J.S. Seneschal, Oblong cuboid illustration
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Crossrefs

Cf. A171608, A000290, A002378, A011379, A045991, A052289.

Programs

Magma

[((-1)^n*(3 -3*n +n^2) -(3 -11*n +9*n^2 -2*n^3))/32: n in [0..60]]; // G. C. Greubel, Apr 08 2022

Maple

A028724:=n->(1/2)*floor(n/2)*floor((n-1)/2)*floor((n-2)/2);
seq(A028724(k), k=0..100); # Wesley Ivan Hurt, Nov 01 2013

Mathematica

Table[(1/2)*Floor[n/2]*Floor[(n-1)/2]*Floor[(n-2)/2], {n,0,100}] (* Wesley Ivan Hurt, Nov 01 2013 *)
Table[(11n-3-9n^2+2n^3+(-1)^n(3-3n+n^2))/32,{n,0,60}] (* Benedict W. J. Irwin, Sep 27 2016 *)
CoefficientList[Series[x^4 (1 + x + x^2)/(x - 1)^4/(x + 1)^3, {x, 0, 60}], x] (* Michael De Vlieger, Sep 27 2016 *)

PARI

{a(n) = (n\2) * ((n-1)\2) * (n\2-1) / 2} /* Michael Somos, Jan 27 2008 */

PARI

{a(n) = if( n<0, n=-1-n; -1, n-=4; 1) * polcoeff( (1 - x^3) / (1 - x)^2 / (1 - x^2)^3 + x*O(x^n), n)} /* Michael Somos, Jan 27 2008 */

PARI

first(n) = Vec(x^4*(1+x+x^2)/(x-1)^4/(x+1)^3 + O(x^(n)), -n) \\ Iain Fox, Nov 18 2017

SageMath

[(1/2)*(n//2)*((n-1)//2)*((n-2)//2) for n in (0..60)] # G. C. Greubel, Apr 08 2022

Formula

G.f.: x^4*(1+x+x^2)/((1-x)^4*(1+x)^3). - Ralf Stephan, Jun 22 2003

Number of tuples [x, y, z, w] of integers such that n = x + y, x >= max(y, z), min(y, z) >= w >= 2. - Michael Somos, Jan 27 2008

Euler transform of length 3 sequence [2, 3, -1]. - Michael Somos, Jan 27 2008

a(3-n) = -a(n). - Michael Somos, Jan 27 2008

a(n) = (-3 + 11*n - 9*n^2 + 2*n^3 + (-1)^n*(3 - 3*n + n^2))/32. - Benedict W. J. Irwin, Sep 27 2016

a(n) = Sum_{i=1..floor((n-1)/2)} i * ( floor((n-1)/2) mod (n-i-1) ). - Wesley Ivan Hurt, Nov 17 2017

E.g.f.: (1/32)*( (3 + 2*x + x^2)*exp(-x) - (1-x)*(3 - x + 2*x^2)*exp(x) ). - G. C. Greubel, Apr 08 2022

From Amiram Eldar, Apr 16 2023: (Start)

Sum_{n>=4} 1/a(n) = 2.

Sum_{n>=4} (-1)^n/a(n) = 2*Pi^2/3 - 6. (End)