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]
FORMULA
G.f.: [1+z+z^2-sqrt(1-2z-5z^2-2z^3+z^4)]/[2z(1+z)^2].
Conjecture: (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);
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jan 06 2005
STATUS
approved