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

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A274115 Number of equivalence classes of Dyck paths of semilength n for the string duu. 9
1, 1, 1, 2, 4, 8, 17, 35, 75, 157, 337, 712, 1529, 3248, 6976, 14869, 31937, 68222, 146536, 313487, 673351, 1441999, 3097326, 6637879, 14257734, 30572032, 65666593, 140860379, 302557585, 649202036, 1394434685, 2992721902, 6428118868, 13798302512, 29637567305, 63626933527, 136665012979 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

a(n+1) is also the number of Dyck meanders of length n, where catastrophes are allowed. A catastrophe is a direct jump from any altitude > 0 to 0, see the Banderier-Wallner article. - Cyril Banderier, May 30 2019

LINKS

Gheorghe Coserea, Table of n, a(n) for n = 0..300

Cyril Banderier, Michael Wallner, Lattice paths with catastrophes, arXiv:1707.01931 [math.CO], 2017.

K. Manes, A. Sapounakis, I. Tasoulas, P. Tsikouras, Equivalence classes of ballot paths modulo strings of length 2 and 3, arXiv:1510.01952 [math.CO], 2015.

FORMULA

A(x) = 1 + x/(1 - x*(1+x)*A000108(x^2)). - Gheorghe Coserea, Jan 06 2017

a(n) = Sum_{k=0..n} (k+1)*Sum_{i=0..(n-k)/2} C(k+1,2*k+2*i-n+3)*C(k+2*i,i))/(k+i+1), n>1, a(0)=1,a(1)=1.  - Vladimir Kruchinin, Feb 14 2019

MATHEMATICA

A[x_] = 1 + x/(1 + ((1 + x)(Sqrt[1 - 4x^2] - 1))/(2x)) + O[x]^40;

CoefficientList[A[x], x] (* Jean-Fran├žois Alcover, Jul 27 2018, after Gheorghe Coserea *)

PROG

(PARI)

seq(N) = {

  my(x='x+O('x^N),

     A000108 = 1+x*Ser(vector(N\2, n, binomial(2*n, n)/(n+1)), 'x));

  Vec(1+x/(1 - x*(1+x)*subst(A000108, 'x, 'x^2)));

};

seq(37)  \\ Gheorghe Coserea, Jan 06 2017

(Maxima)

a(n):=if n<2 then 1 else sum((k+1)*sum((binomial(k+1, 2*k+2*i-n+3)*binomial(k+2*i, i))/(k+i+1), i, 0, (n-k)/2), k, 0, n); /* Vladimir Kruchinin, Feb 14 2019 */

CROSSREFS

Cf. A274110, A274111, A274112, A274113, A274114.

Sequence in context: A058520 A127680 A136750 * A097107 A098083 A182900

Adjacent sequences:  A274112 A274113 A274114 * A274116 A274117 A274118

KEYWORD

nonn,walk

AUTHOR

N. J. A. Sloane, Jun 17 2016

EXTENSIONS

a(0)=1 prepended and more terms from Gheorghe Coserea, Jan 06 2017

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.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 09:33 EST 2019. Contains 329843 sequences. (Running on oeis4.)