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-François 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
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