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A128737 Number of LDU's in all skew Dyck paths of semilength n. 1
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 (list; graph; refs; listen; history; text; internal format)
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: -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

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Last modified April 25 12:04 EDT 2019. Contains 322456 sequences. (Running on oeis4.)