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A129155 Number of skew Dyck paths of semilength n that have no primitive Dyck factors.

%I A129155 #12 Mar 21 2017 04:20:21
%S A129155 1,0,1,4,15,59,241,1011,4326,18797,82685,367410,1646494,7432270,
%T A129155 33761322,154213566,707882503,3263713148,15107319268,70182332975,
%U A129155 327111450097,1529226524057,7168880978609,33693179852563
%N A129155 Number of skew Dyck paths of semilength n that have no primitive Dyck factors.
%C A129155 A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. A primitive Dyck factor is a subpath of the form UPD that starts on the x-axis, P being a Dyck path.
%H A129155 G. C. Greubel, <a href="/A129155/b129155.txt">Table of n, a(n) for n = 0..1000</a>
%H A129155 E. Deutsch, E. Munarini, S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
%F A129155 a(n) = A129154(n,0).
%F A129155 G.f.: (3-3*z-sqrt(1-6*z+5*z^2))/(2+z-sqrt(1-4*z)+sqrt(1-6*z+5*z^2)).
%F A129155 a(n) ~ (475 + 697*sqrt(5)) * 5^n / (3364*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 20 2014
%e A129155 a(3)=4 because we have UUUDLD, UUDUDL, UUUDDL and UUUDLL.
%p A129155 G:=(3-3*z-sqrt(1-6*z+5*z^2))/(2+z-sqrt(1-4*z)+sqrt(1-6*z+5*z^2)): Gser:=series(G,z=0,32): seq(coeff(Gser,z,n),n=0..28);
%t A129155 CoefficientList[Series[(3-3*x-Sqrt[1-6*x+5*x^2])/(2+x-Sqrt[1-4*x]+Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o A129155 (PARI) z='z+O('z^50); Vec((3-3*z-sqrt(1-6*z+5*z^2))/(2+z-sqrt(1-4*z)+sqrt(1-6*z+5*z^2))) \\ _G. C. Greubel_, Mar 20 2017
%Y A129155 Cf. A129154, A129157.
%K A129155 nonn
%O A129155 0,4
%A A129155 _Emeric Deutsch_, Apr 02 2007