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a(n) = (1/2)*floor(n/2)*floor((n-1)/2)*floor((n-2)/2).
8

%I #62 Apr 16 2023 03:16:25

%S 0,0,0,0,1,2,6,9,18,24,40,50,75,90,126,147,196,224,288,324,405,450,

%T 550,605,726,792,936,1014,1183,1274,1470,1575,1800,1920,2176,2312,

%U 2601,2754,3078,3249,3610,3800,4200,4410,4851,5082,5566,5819,6348,6624,7200

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

%C 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

%C Starting with "1" = triangle A171608 * the triangular numbers. - _Gary W. Adamson_, Dec 12 2009

%C 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

%D 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.

%H Iain Fox, <a href="/A028724/b028724.txt">Table of n, a(n) for n = 0..10000</a>

%H Paul Barry, <a href="http://arxiv.org/abs/1205.2565">On sequences with {-1, 0, 1} Hankel transforms</a>, arXiv preprint arXiv:1205.2565 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 18 2012

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).

%F G.f.: x^4*(1+x+x^2)/((1-x)^4*(1+x)^3). - _Ralf Stephan_, Jun 22 2003

%F 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

%F Euler transform of length 3 sequence [2, 3, -1]. - _Michael Somos_, Jan 27 2008

%F a(3-n) = -a(n). - _Michael Somos_, Jan 27 2008

%F 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

%F a(n) = Sum_{i=1..floor((n-1)/2)} i * ( floor((n-1)/2) mod (n-i-1) ). - _Wesley Ivan Hurt_, Nov 17 2017

%F 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

%F From _Amiram Eldar_, Apr 16 2023: (Start)

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

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

%e 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.

%p A028724:=n->(1/2)*floor(n/2)*floor((n-1)/2)*floor((n-2)/2);

%p seq(A028724(k), k=0..100); # _Wesley Ivan Hurt_, Nov 01 2013

%t Table[(1/2)*Floor[n/2]*Floor[(n-1)/2]*Floor[(n-2)/2], {n,0,100}] (* _Wesley Ivan Hurt_, Nov 01 2013 *)

%t Table[(11n-3-9n^2+2n^3+(-1)^n(3-3n+n^2))/32,{n,0,60}] (* _Benedict W. J. Irwin_, Sep 27 2016 *)

%t CoefficientList[Series[x^4 (1 + x + x^2)/(x - 1)^4/(x + 1)^3, {x, 0, 60}], x] (* _Michael De Vlieger_, Sep 27 2016 *)

%o (PARI) {a(n) = (n\2) * ((n-1)\2) * (n\2-1) / 2} /* _Michael Somos_, Jan 27 2008 */

%o (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 */

%o (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

%o (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

%o (SageMath) [(1/2)*(n//2)*((n-1)//2)*((n-2)//2) for n in (0..60)] # _G. C. Greubel_, Apr 08 2022

%Y Cf. A171608.

%K nonn,easy

%O 0,6

%A _N. J. A. Sloane_