This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A128743 Number of UU's (i.e., doublerises) in all skew Dyck paths of semilength n. 2
 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 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. 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-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 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 *) PROG (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 CROSSREFS Cf. A128718. Sequence in context: A097977 A136780 A301944 * A218184 A264735 A289926 Adjacent sequences:  A128740 A128741 A128742 * A128744 A128745 A128746 KEYWORD nonn AUTHOR Emeric Deutsch, Mar 30 2007 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 27 04:05 EDT 2019. Contains 323597 sequences. (Running on oeis4.)