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

 

Logo

The submissions stack has been unacceptably high for several months now. Please voluntarily restrict your submissions and please help with the editing. (We don't want to have to impose further limits.)

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A129156 Number of primitive Dyck factors 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 the path is defined to be the number of its steps. A primitive Dyck factor is a subpath of the form UPD that starts on the x-axis, P being a Dyck path. 2
0, 1, 3, 10, 36, 136, 532, 2139, 8796, 36859, 156946, 677514, 2959669, 13063493, 58184838, 261230814, 1181144792, 5374078726, 24588562675, 113067256235, 522270436044, 2422244159067, 11275548912967, 52663412854571 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n)=Sum(k*A129154(n,k),k=0..n). a(n)=A128742(n)-A129158(n).

LINKS

Table of n, a(n) for n=0..23.

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

FORMULA

G.f.=[3-3z-sqrt(1-6z+5z^2)][1-sqrt(1-4z)]/[1+z+sqrt(1-6*z+5*z^2)]^2.

a(n) ~ (5-sqrt(5)) * 5^(n+3/2) / (36*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

EXAMPLE

a(2)=3 because in all skew Dyck paths of semilength 3, namely (UD)(UD), (UUDD) and UUDL, we have altogether 3 primitive Dyck factors (shown between parentheses).

MAPLE

G:=(3-3*z-sqrt(1-6*z+5*z^2))*(1-sqrt(1-4*z))/(1+z+sqrt(1-6*z+5*z^2))^2: Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27);

MATHEMATICA

CoefficientList[Series[(3-3*x-Sqrt[1-6*x+5*x^2])*(1-Sqrt[1-4*x])/ (1+x+Sqrt[1-6*x+5*x^2])^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

CROSSREFS

Cf. A129154, A129157, A129158.

Sequence in context: A007582 A026854 A136576 * A171753 A002212 A149041

Adjacent sequences:  A129153 A129154 A129155 * A129157 A129158 A129159

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Apr 02 2007

STATUS

approved

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

Content is available under The OEIS End-User License Agreement .

Last modified August 28 22:18 EDT 2015. Contains 261164 sequences.