A128743 Number of UU's (i.e., doublerises) in all skew Dyck paths of semilength n.

0, 0, 2, 13, 69, 346, 1700, 8286, 40264, 195488, 949302, 4613025, 22436997, 109240038, 532410060, 2597468685, 12684628125, 62002335160, 303332650190, 1485213237135, 7277719953415, 35687662907750, 175120787451540

Offset: 0

Author

Emeric Deutsch, Mar 30 2007

Keywords

• nonn

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 its steps.

Examples

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

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.

Crossrefs

Cf. A128718.

Programs

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 *)

PARI

z='z+O('z^50); concat([0,0], Vec((1-4*z+z^2+(z-1)*sqrt(1-6*z+5*z^2))/(2*z*sqrt(1-6*z+5*z^2)))) \\ G. C. Greubel, Mar 20 2017

Formula

a(n) = Sum_{k=0..n-1} k*A128718(n,k).

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

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

Conjecture: (n+1)*(n-2)^2*a(n) -(n-1)*(6*n^2-15*n+4)*a(n-1) +5*(n-2)*(n-1)^2*a(n-2)=0. - R. J. Mathar, Jun 17 2016

Conjecture verified using the differential equation 4*g(z)+(20*z^3+2*z^2-2*z)*g'(z)+(25*z^4-15*z^3)*g''(z)+(5*z^5-6*z^4+z^3)*g'''(z)=0 satisfied by the G.f. - Robert Israel, Dec 25 2017