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A128743 Number of UU's (i.e. doublerises) in all skew Dyck paths of semilength n. 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 its steps. 1
0, 0, 2, 13, 69, 346, 1700, 8286, 40264, 195488, 949302, 4613025, 22436997, 109240038, 532410060, 2597468685, 12684628125, 62002335160, 303332650190, 1485213237135, 7277719953415, 35687662907750, 175120787451540 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n)=Sum[k*A128718(n,k), k=0..n-1].

LINKS

Table of n, a(n) for n=0..22.

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

FORMULA

G.f.=[1-4z+z^2+(z-1)sqrt(1-6z+5z^2)]/[2z*sqrt(1-6z+5z^2)].

a(n) ~ 3*5^(n-1/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014

EXAMPLE

a(2)=2 because the paths of semilength 2 are UDUD, UUDD and UUDL, having altogether 2 UU's.

MAPLE

G:=(1-4*z+z^2+(z-1)*sqrt(1-6*z+5*z^2))/2/z/sqrt(1-6*z+5*z^2): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..25);

MATHEMATICA

CoefficientList[Series[(1-4*x+x^2+(x-1)*Sqrt[1-6*x+5*x^2])/2/x/Sqrt[1-6*x+5*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

CROSSREFS

Cf. A128718.

Sequence in context: A038144 A097977 A136780 * A218184 A188676 A097349

Adjacent sequences:  A128740 A128741 A128742 * A128744 A128745 A128746

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Mar 30 2007

STATUS

approved

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Last modified November 23 19:50 EST 2014. Contains 249865 sequences.