OFFSET
0,4
COMMENTS
Number of Łukasiewicz paths of length n having no level steps at an even level. A Łukasiewicz path of length n is a path in the first quadrant from (0,0) to (n,0) using rise steps (1,k) for any positive integer k, level steps (1,0) and fall steps (1,-1) (see R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999, p. 223, Exercise 6.19w; the integers are the slopes of the steps). Example: a(3)=2 because we have UHD and U(2)DD, where U=(1,1), H=(1,0), D=(1,-1) and U(2)=(1,2). a(n)=A102404(n,0).
Number of Dyck n-paths with no descent of length 1 following an ascent of length 1. [David Scambler, May 11 2012]
LINKS
FORMULA
G.f.: (1+z+z^2 - sqrt(1-2*z-5*z^2-2*z^3+z^4))/(2*z*(1+z)^2).
(n+1)*a(n) -(n-3)*a(n-1) -(7*n-9)*a(n-2) -(7*n-12)*a(n-3) -n*a(n-4) +(n-4)*a(n-5) = 0. - R. J. Mathar, Jan 04 2017
EXAMPLE
a(3) = 2 because we have UUDUDD and UUUDDD, having no ascents of length 1 that start at an even level.
MAPLE
G:=(1+z+z^2-sqrt(1-2*z-5*z^2-2*z^3+z^4))/2/z/(1+z)^2: Gser:=series(G, z=0, 32): 1, seq(coeff(Gser, z^n), n=1..29);
MATHEMATICA
CoefficientList[Series[(1+x+x^2 -Sqrt[1-2*x-5*x^2-2*x^3+x^4])/(2*x*(1+x)^2), {x, 0, 40}], x] (* G. C. Greubel, Oct 31 2024 *)
PROG
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1+x+x^2 -Sqrt(1-2*x-5*x^2-2*x^3+x^4))/(2*x*(1+x)^2) )); // G. C. Greubel, Oct 31 2024
(SageMath)
def A102406_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1+x+x^2 -sqrt(1-2*x-5*x^2-2*x^3+x^4))/(2*x*(1+x)^2) ).list()
A102406_list(30) # G. C. Greubel, Oct 31 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jan 06 2005
STATUS
approved