A128737 Number of LDU's in all skew Dyck paths of semilength n.

0, 0, 0, 0, 1, 10, 69, 412, 2291, 12244, 63886, 328256, 1669363, 8429384, 42349096, 211982828, 1058244079, 5272285552, 26227527576, 130323237088, 647013004499, 3210128312122, 15919166804461, 78915323039268, 391100149306301

Offset

0,6

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=0..24.

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

Formula

a(n) = Sum_{k>=0} k*A128735(n,k).

G.f.: z(g-1)^3/(4g - 2zg - 6zg^2 - 3 + 3*z), where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z).

Conjecture D-finite with recurrence: -10*(n+1)*(n-4)*a(n) +(73*n^2-273*n+140)*a(n-1) +(-132*n^2+641*n-734) *a(n-2) +(n-3)*(89*n-269)*a(n-3) -20*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jun 17 2016

Example

a(4)=1 because among the 36 (=A002212(4)) skew Dyck paths of semilength 4 only UUUDLDUD has a LDU.

Maple

g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: ser:=series(z*(g-1)^3/(4*g-2*z*g-6*z*g^2-3+3*z), z=0, 30): seq(coeff(ser, z, n), n=0..27);

Crossrefs

Cf. A128735.

Sequence in context: A081280 A038806 A016273 * A320228 A130548 A160662

Adjacent sequences: A128734 A128735 A128736 * A128738 A128739 A128740

Keyword

nonn

Author

Emeric Deutsch, Mar 31 2007

Status

approved