login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A191391 Number of horizontal segments in all dispersed Dyck paths of length n (i.e., in all Motzkin paths of length n with no (1,0)-steps at positive heights; a horizontal segment is a maximal sequence of consecutive (1,0)-steps). 2
0, 1, 1, 3, 5, 12, 22, 49, 93, 200, 386, 814, 1586, 3304, 6476, 13381, 26333, 54096, 106762, 218386, 431910, 880616, 1744436, 3547658, 7036530, 14281072, 28354132, 57451164, 114159428, 230993296, 459312152, 928319149, 1846943453, 3729244576, 7423131482, 14975907754 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..35.

FORMULA

a(n) = Sum_{k>=0} k*A191390(n,k).

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

a(0)=0 and a(n)=2^(n-1)-C(n-1,floor(n/2)-1) for n>=1. [Joerg Arndt, Aug 07 2012, aeb]

D-finite with recurrence (n+1)*a(n) +(-3*n-1)*a(n-1) +2*(-n+3)*a(n-2) +4*(3*n-8)*a(n-3) +8*(-n+4)*a(n-4)=0. - R. J. Mathar, Jul 24 2022

EXAMPLE

a(4)=5 because in (HHHH), (HH)UD, (H)UD(H), UD(HH), UDUD, and UUDD we have a total of 1+1+2+1+0+0=5 horizontal segments (shown between parentheses).

MAPLE

g := 4*z*(1-z)/(1-2*z+sqrt(1-4*z^2))^2: gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 35);

CROSSREFS

Cf. A191390.

First differences of A045621.

Sequence in context: A034763 A183921 A177143 * A121482 A013498 A161624

Adjacent sequences: A191388 A191389 A191390 * A191392 A191393 A191394

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jun 03 2011

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 21:40 EST 2022. Contains 358594 sequences. (Running on oeis4.)