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

Jean-Luc Baril and Sergey Kirgizov, Bijections from Dyck and Motzkin meanders with catastrophes to pattern avoiding Dyck paths, arXiv:2104.01186 [math.CO], 2021.

K. Manes, A. Sapounakis, I. Tasoulas, and 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

D-finite with recurrence (-n+1)*a(n) +2*a(n-1) +7*(n-3)*a(n-2) +3*(n-5)*a(n-3) +(-11*n+53)*a(n-4) +4*(-3*n+16)*a(n-5) +4*(-n+6)*a(n-6)=0. - R. J. Mathar, Sep 27 2020

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. A000108, 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

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Last modified October 2 23:51 EDT 2022. Contains 357230 sequences. (Running on oeis4.)