A129162 Sum of heights of all skew Dyck paths of semilength n.

1, 5, 23, 103, 462, 2086, 9493, 43521, 200855, 932429, 4350995, 20395349, 95987113, 453354623, 2148027772, 10206485598, 48621125308, 232156538970, 1110842790406, 5325499426116, 25576096186920, 123030491611330

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

Table of n, a(n) for n=1..22.

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} k*A129161(n,k).

Example

a(2)=5 because the paths UDUD, UUDD and UUDL have heights 1, 2 and 2, respectively.

Maple

H[0]:=1: for k from 1 to 32 do H[k]:=simplify((1+z*H[k-1]-z)/(1-z*H[k-1])) od: for k from 1 to 32 do h[k]:=factor(simplify(H[k]-H[k-1])) od: for k from 1 to 32 do hser[k]:=series(h[k], z=0, 30) od: T:=(n, k)->coeff(hser[k], z, n): seq(add(k*T(n, k), k=1..n), n=1..25);

Crossrefs

Cf. A129161.

Sequence in context: A289803 A102285 A218985 * A377956 A167660 A290924

Adjacent sequences: A129159 A129160 A129161 * A129163 A129164 A129165

Keyword

nonn

Author

Emeric Deutsch, Apr 03 2007

Status

approved