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%I #20 Jan 04 2022 03:00:27
%S 1,0,2,6,22,84,334,1368,5734,24480,106086,465462,2063658,9231084,
%T 41610162,188820726,861891478,3954732384,18230522422,84390187986,
%U 392120098258,1828220666844,8550445900442,40103716079436
%N Number of skew Dyck paths of semilength n having no peaks at level 1.
%C 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 a path is defined to be the number of steps in it.
%H G. C. Greubel, <a href="/A128723/b128723.txt">Table of n, a(n) for n = 0..1000</a>
%H E. Deutsch, E. Munarini and 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.
%H Helmut Prodinger, <a href="https://arxiv.org/abs/2201.00640">Skew Dyck paths having no peaks at level 1</a>, arXiv:2201.00640 [math.CO], 2022.
%F a(n) = A128722(n,0).
%F a(n) = 2*A117641(n) for n>=1.
%F G.f.: (3-3*z-sqrt(1-6*z+5*z^2))/(1+3*z+sqrt(1-6*z+5*z^2)).
%F a(n) ~ 5^(n+3/2)/(16*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 20 2014
%F Conjecture: +3*(n+1)*a(n) +(-17*n+10)*a(n-1) +9*(n-3)*a(n-2) +5*(n-2)*a(n-3)=0. - _R. J. Mathar_, Jun 17 2016
%e a(3)=6 because we have UUDUDD, UUUDDD, UUUDLD, UUDUDL, UUUDDL and UUUDLL.
%p G:=(3-3*z-sqrt(1-6*z+5*z^2))/(1+3*z+sqrt(1-6*z+5*z^2)): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..27);
%t CoefficientList[Series[(3-3*x-Sqrt[1-6*x+5*x^2])/(1+3*x+Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o (PARI) my(z='x+O('z^50)); Vec((3-3*z-sqrt(1-6*z+5*z^2)) /(1+3*z +sqrt(1-6*z+5*z^2))) \\ _G. C. Greubel_, Mar 19 2017
%Y Cf. A117641, A128722.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Mar 31 2007
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