%I #11 Jan 04 2017 14:28:58

%S 1,0,1,2,5,14,39,114,339,1028,3163,9852,31005,98436,314901,1014070,

%T 3284657,10694314,34979667,114887846,378750951,1252865288,4157150327,

%U 13832926200,46148704121,154327715592,517236429545,1737102081962,5845077156189,19702791805126

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

%C 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).

%C Number of Dyck n-paths with no descent of length 1 following an ascent of length 1. [_David Scambler_, May 11 2012]

%H <a href="/index/Lu#Lukasiewicz">Index entries for sequences related to Łukasiewicz</a>

%F G.f.: [1+z+z^2-sqrt(1-2z-5z^2-2z^3+z^4)]/[2z(1+z)^2].

%F 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

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

%p 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);

%Y Cf. A102404, A102407.

%K nonn

%O 0,4

%A _Emeric Deutsch_, Jan 06 2005