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

%I A102406 #16 Oct 31 2024 12:23:36
%S A102406 1,0,1,2,5,14,39,114,339,1028,3163,9852,31005,98436,314901,1014070,
%T A102406 3284657,10694314,34979667,114887846,378750951,1252865288,4157150327,
%U A102406 13832926200,46148704121,154327715592,517236429545,1737102081962,5845077156189,19702791805126
%N A102406 Number of Dyck paths of semilength n having no ascents of length 1 that start at an even level.
%C A102406 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 A102406 Number of Dyck n-paths with no descent of length 1 following an ascent of length 1. [_David Scambler_, May 11 2012]
%H A102406 G. C. Greubel, <a href="/A102406/b102406.txt">Table of n, a(n) for n = 0..1000</a>
%H A102406 <a href="/index/Lu#Lukasiewicz">Index entries for sequences related to Łukasiewicz</a>
%F A102406 G.f.: (1+z+z^2 - sqrt(1-2*z-5*z^2-2*z^3+z^4))/(2*z*(1+z)^2).
%F A102406 (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 A102406 a(3) = 2 because we have UUDUDD and UUUDDD, having no ascents of length 1 that start at an even level.
%p A102406 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);
%t A102406 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 *)
%o A102406 (Magma)
%o A102406 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A102406 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
%o A102406 (SageMath)
%o A102406 def A102406_list(prec):
%o A102406     P.<x> = PowerSeriesRing(ZZ, prec)
%o A102406     return P( (1+x+x^2 -sqrt(1-2*x-5*x^2-2*x^3+x^4))/(2*x*(1+x)^2) ).list()
%o A102406 A102406_list(30) # _G. C. Greubel_, Oct 31 2024
%Y A102406 Cf. A102404, A102407.
%K A102406 nonn
%O A102406 0,4
%A A102406 _Emeric Deutsch_, Jan 06 2005