A128746 Height of the last peak summed over all skew Dyck paths of semilength n.

1, 5, 22, 94, 401, 1723, 7475, 32749, 144803, 645627, 2900256, 13115820, 59669295, 272918415, 1254314310, 5789850730, 26831078075, 124785337255, 582247766810, 2724905891890, 12787603121195, 60162698218325, 283715348775727

Offset

1,2

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, Table of n, a(n) for n = 1..1000

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

Formula

a(n) = Sum_{k=1,..,n} A128745(n,k).

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

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

Conjecture: -(n+2)*(n-1)*a(n) +(6*n^2-3*n+2)*a(n-1) -5*n*(n-2)*a(n-2)=0. - R. J. Mathar, Aug 08 2015

Example

a(2)=5 because the skew Dyck paths of semilength 2 are UD(UD), U(UD)D and U(UD)L and their last peaks (shown between parentheses) have heights 1, 2 and 2, respectively.

Maple

G:=2*z*(1+z+sqrt(1-6*z+5*z^2))/(1-3*z+sqrt(1-6*z+5*z^2))^2: Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=1..27);

Mathematica

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

Prog

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

Crossrefs

Cf. A128745.

Sequence in context: A026672 A049652 A026877 * A049675 A053154 A141222

Adjacent sequences: A128743 A128744 A128745 * A128747 A128748 A128749

Keyword

nonn

Author

Emeric Deutsch, Mar 31 2007

Status

approved