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A354539 Number of decorated Dyck paths of length n without peaks at level 1 ending at arbitrary levels.

%I A354539 #11 Mar 02 2023 08:33:20
%S A354539 1,1,1,2,5,8,18,31,71,126,290,527,1218,2253,5223,9796,22763,43170,
%T A354539 100502,192347,448476,864887,2019121,3919162,9159252,17877619,
%U A354539 41819003,82021628,192015633
%N A354539 Number of decorated Dyck paths of length n without peaks at level 1 ending at arbitrary levels.
%H A354539 H. Prodinger, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Prodinger/prod36.html">Skew Dyck paths having no peaks at level 1</a>, JIS 25 (2022) # 22.1.16, section 2.3.
%F A354539 G.f.: (-2*z^5-3*z^4+z^3-5*z^2-3*z+4-(z^2+3*z+4)*sqrt(1-6*z^2+5*z^4))/2/z/(3+z^2)/(z^2+2*z-1) .
%F A354539 D-finite with recurrence 12*(n+1)*a(n) +3*(-5*n-11)*a(n-1) +5*(-19*n+29)*a(n-2) +14*(5*n-4)*a(n-3) +2*(93*n-356)*a(n-4) +2*(20*n-81)*a(n-5) +2*(-22*n+217)*a(n-6) +2*(-35*n+268)*a(n-7) +2*(-27*n+182)*a(n-8) +5*(-5*n+39)*a(n-9) +5*(-n+9)*a(n-10)=0.
%p A354539 g := (-2*z^5-3*z^4+z^3-5*z^2-3*z+4-(z^2+3*z+4)*sqrt(1-6*z^2+5*z^4))/2/z/(3+z^2)/(z^2+2*z-1) ;
%p A354539 taylor(%,z=0,30) ;
%p A354539 gfun[seriestolist](%) ;
%Y A354539 Cf. A128723 (ending at level 0).
%K A354539 nonn
%O A354539 0,4
%A A354539 _R. J. Mathar_, Aug 17 2022