A129173 Total area below all skew Dyck paths of semilength n.

0, 1, 9, 58, 336, 1853, 9945, 52487, 273939, 1418567, 7303791, 37441560, 191287254, 974642943, 4955123955, 25146686730, 127424717400, 644873878895, 3260055588615, 16465301636090, 83092583965020, 419031686115875

Offset: 0

Author

Emeric Deutsch, Apr 09 2007

Keywords

• nonn

Comments

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.

Examples

			a(2)=9 because the areas below the skew Dyck paths UDUD, UUDD and UUDL are 2, 4 and 3, respectively.
		

Links

• G. C. Greubel and Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (terms 0..300 from Vincenzo Librandi)
• E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
• Emeric Deutsch, Emanuele Munarini and Simone Rinaldi, Skew Dyck paths, area, and superdiagonal bargraphs, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562, Table 1.

Crossrefs

Cf. A002212, A129172.

Programs

Maple

a[0]:=1: a[1]:=1: a[2]:=9: for n from 3 to 25 do a[n]:=((11*n^2-20*n-6)*a[n-1]-5*(7*n^2-19*n+7)*a[n-2]+25*(n-1)*(n-3)*a[n-3])/(n+1)/(n-2) od: seq(a[n],n=0..25);

Mathematica

CoefficientList[Series[(1+x)*(1-3*x-Sqrt[1-6*x+5*x^2])/(2*x*(1-5*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

PARI

z='z +O('z^25); concat([0], Vec((1+z)*(1-3*z-sqrt(1-6*z+5*z^2))/(2*z*(1-5*z)))) \\ G. C. Greubel, Feb 10 2017

Formula

a(n) = Sum_{k=0,..,n^2} k*A129172(n,k).

a(n) - 5*a(n-1) = A002212(n) + A002212(n-1).

G.f.: (1+z)*(1-3*z-sqrt(1-6*z+5*z^2))/(2*z*(1-5*z)).

(n+1)(n-2)a(n)-(11n^2-20n-6)a(n-1)+5(7n^2-19n+7)a(n-2)-25(n-1)(n-3)a(n-3) = 0.

a(n) ~ 6*5^(n-1)*(1-sqrt(5)/sqrt(Pi*n)) . - Vaclav Kotesovec, Oct 20 2012