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

0,3

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.

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.

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A002212, A129172.

Sequence in context: A026377 A016209 A196920 * A271271 A055423 A322207

Adjacent sequences: A129170 A129171 A129172 * A129174 A129175 A129176

Keyword

nonn

Author

Emeric Deutsch, Apr 09 2007

Status

approved