A128734 Number of LD's in all skew Dyck paths of semilength n.

0, 0, 0, 1, 8, 48, 261, 1358, 6907, 34684, 172850, 857389, 4240442, 20933422, 103221134, 508623877, 2505298946, 12338127048, 60761615904, 299256606347, 1474086307696, 7262524940428, 35789196572489, 176410731649052

Offset

0,5

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 a path is defined to be the number of its steps.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} k*A128733(n,k), for n >= 1.

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

a(n) ~ 5^(n-1/2)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014

Conjecture: +2*(3*n-20)*(n-3)*(n+1)*a(n) +(-39*n^3+362*n^2-739*n+300)*a(n-1) +8*(n-2) *(6*n^2-49*n+75) *a(n-2) -5*(n-2)*(n-3)*(3*n-17)*a(n-3)=0. - R. J. Mathar, Jun 17 2016

Example

a(4)=8 because we have UDUUUD(LD), UUUD(LD)UD, UUDUUD(LD), UUUUD(LD)D, UUUDUD(LD), UUUUDD(LD), UUUUDL(LD) and UUUUD(LD)L (the LD's are shown between parentheses; the other 28 skew Dyck paths of semilength 4 contain no LD).

Maple

g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=z^2*g^2*(g-1)/(1-3*z+2*z^2-3*z^2*g^2): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..26);

Mathematica

CoefficientList[Series[x^2*((1-x-Sqrt[1-6*x+5*x^2])/2/x)^2*((1-x-Sqrt[1-6*x+5*x^2])/2/x-1)/(1-3*x+2*x^2-3*x^2*((1-x-Sqrt[1-6*x+5*x^2])/2/x)^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

Prog

(PARI) x='x+O('x^50); concat([0, 0, 0], Vec(x^2*((1-x-sqrt(1-6*x+5*x^2))/2/x)^2*((1-x-sqrt(1-6*x+5*x^2))/2/x-1)/(1-3*x+2*x^2-3*x^2*((1-x-sqrt(1-6*x+5*x^2))/2/x)^2))) \\ G. C. Greubel, Mar 20 2017

Crossrefs

Cf. A128733.

Sequence in context: A285063 A026761 A026706 * A006321 A371620 A295047

Adjacent sequences: A128731 A128732 A128733 * A128735 A128736 A128737

Keyword

nonn

Author

Emeric Deutsch, Mar 31 2007

Status

approved