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A128726 Number of LL's in all skew Dyck paths of semilength n. 2
0, 0, 0, 1, 7, 38, 192, 946, 4616, 22440, 108964, 529133, 2571079, 12504038, 60872038, 296641049, 1447054867, 7065841496, 34534088328, 168933369259, 827073303197, 4052396628306, 19870029768028, 97495408609784 (list; graph; refs; listen; history; text; internal format)
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 steps in it.

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..n-2} k*A128724(n,k).

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

a(n) ~ 5^(n-1/2)/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014

Conjecture: +2*n*(n-3)^2*a(n) +(-13*n^3+84*n^2-167*n+100)*a(n-1) +(n-2)*(16*n^2-77*n+85)*a(n-2) -5*(n-3)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Jun 17 2016

EXAMPLE

a(4)=7 because we have UDUUUDLL, UUUUDLLD, UUDUUDLL, UUUDUDLL, UUUUDDLL and UUUUDLLL (last path has two LL's).

MAPLE

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

MATHEMATICA

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

CROSSREFS

Cf. A128724.

Sequence in context: A037696 A026895 A037605 * A346395 A055146 A014827

Adjacent sequences:  A128723 A128724 A128725 * A128727 A128728 A128729

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Mar 31 2007

STATUS

approved

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Last modified July 29 12:41 EDT 2021. Contains 346346 sequences. (Running on oeis4.)