A102406 Number of Dyck paths of semilength n having no ascents of length 1 that start at an even level.

1, 0, 1, 2, 5, 14, 39, 114, 339, 1028, 3163, 9852, 31005, 98436, 314901, 1014070, 3284657, 10694314, 34979667, 114887846, 378750951, 1252865288, 4157150327, 13832926200, 46148704121, 154327715592, 517236429545, 1737102081962, 5845077156189, 19702791805126

Offset: 0

Author

Emeric Deutsch, Jan 06 2005

Keywords

• nonn

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]

Examples

			a(3) = 2 because we have UUDUDD and UUUDDD, having no ascents of length 1 that start at an even level.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for sequences related to Łukasiewicz

Crossrefs

Cf. A102404, A102407.

Programs

Magma

R:=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

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 *)

SageMath

def A102406_list(prec):
    P. = 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

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