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!)
A135336 Number of Dyck paths of semilength n with no UUDU's starting at level 0. 1
1, 1, 2, 4, 10, 28, 85, 271, 893, 3013, 10351, 36075, 127219, 453097, 1627378, 5887660, 21436354, 78484402, 288779728, 1067263660, 3960081904, 14746806292, 55094725918, 206450572930, 775724723086, 2922060848734 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Column 0 of A135330. Partial sums of the Fine sequence 1,0,1,2,6,18,... (A000957 without the first term). - Emeric Deutsch, Dec 14 2007

LINKS

Table of n, a(n) for n=0..25.

A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

FORMULA

a(n) = Sum[(-1)^j*(3j+1)binom(2n-3j,n),j=0..floor(n/3)]/(n+1). G.f.: C/(1+z^3*C^3)=C/[(1-z)(1+zC)], where C=[1-sqrt(1-4z)]/(2z) is the g.f. of the Catalan numbers (A000108). - Emeric Deutsch, Dec 14 2007

a(n) ~ 4^(n+2)/(27*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

EXAMPLE

a(3)=4 because among the 5 (=A000108(3)) Dyck paths of semilength 3 only UUDUDD does not qualify.

MAPLE

a:=proc(n) options operator, arrow: (sum((-1)^j*(3*j+1)*binomial(2*n-3*j, n), j =0..floor((1/3)*n)))/(n+1) end proc: seq(a(n), n=0..25); # Emeric Deutsch, Dec 14 2007

MATHEMATICA

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

CROSSREFS

Cf. A000108, A135330, A000957.

Sequence in context: A068875 A262543 A192574 * A149825 A149826 A149827

Adjacent sequences:  A135333 A135334 A135335 * A135337 A135338 A135339

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Dec 07 2007

EXTENSIONS

Edited and extended by Emeric Deutsch, Dec 14 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 9 21:41 EST 2016. Contains 278987 sequences.