login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

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. 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

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

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 A264735 A188676

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 4 17:24 EST 2016. Contains 278755 sequences.