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%I #18 Nov 14 2023 13:43:35
%S 1,1,2,5,13,36,105,317,982,3105,9981,32520,107157,356481,1195662,
%T 4038909,13728369,46919812,161143157,555857157,1924956954,6689953057,
%U 23325404153,81567552320,286009944649,1005371062561,3542175587306
%N Number of Dyck paths of semilength n having no ascents of length 2 that start at an odd level.
%C Column 0 of A114463.
%H G. C. Greubel, <a href="/A114465/b114465.txt">Table of n, a(n) for n = 0..1000</a>
%H Jean-Luc Baril and Paul Barry, <a href="https://arxiv.org/abs/2212.12404">Two kinds of partial Motzkin paths with air pockets</a>, arXiv:2212.12404 [math.CO], 2022.
%H Jean-Luc Baril, Daniela Colmenares, José L. Ramírez, Emmanuel D. Silva, Lina M. Simbaqueba, and Diana A. Toquica, <a href="http://jl.baril.u-bourgogne.fr/bacorasisito.pdf">Consecutive pattern-avoidance in Catalan words according to the last symbol</a>, Univ. Bourgogne (France 2023).
%H Jean-Luc Baril, Rigoberto Flórez, and José L. Ramírez, <a href="http://jl.baril.u-bourgogne.fr/symasympyramid.pdf">Counting symmetric and asymmetric peaks in motzkin paths with air pockets</a>, Univ. Bourgogne (France, 2023).
%F G.f.: [1 - z^2 - sqrt((1+z^2)*(1-4z+z^2))]/[2*z*(1-z+z^2)].
%F (n+1)*a(n) = (5*n-1)*a(n-1) - (7*n-5)*a(n-2) + 10*(n-2)*a(n-3) - (7*n-23)*a(n-4) + (5*n-19)*a(n-5) - (n-5)*a(n-6). - _Vaclav Kotesovec_, Mar 20 2014
%F a(n) ~ sqrt(24+14*sqrt(3)) * (2+sqrt(3))^n / (6 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Mar 20 2014
%e a(4)=13 because among the 14 Dyck paths of semilength 4 only UUD(UU)DDD has an ascent of length 2 that starts at an odd level (shown between parentheses).
%p g:=-1/2/z/(1+z^2-z)*(z^2-1+sqrt((z^2+1)*(z^2-4*z+1))): gser:=series(g,z=0,33): 1,seq(coeff(gser,z^n),n=1..30);
%t CoefficientList[Series[(1-x^2-Sqrt[(1+x^2)*(1-4*x+x^2)])/(2*x*(1-x+x^2)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o (PARI) Vec((1 - x^2 - sqrt((1+x^2)*(1-4*x+x^2)))/(2*x*(1-x+x^2)) + O(x^50)) \\ _G. C. Greubel_, Jan 28 2017
%Y Cf. A114463, A114462, A114464.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Nov 29 2005
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