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A105641 Number of hill-free Dyck paths of semilength n, having no UUDD's, where U=(1,1) and D=(1,-1) (a hill in a Dyck path is a peak at level 1).

%I A105641 #10 Jul 24 2022 11:25:05
%S A105641 0,1,2,5,14,39,111,322,947,2818,8470,25677,78420,241061,745265,
%T A105641 2315794,7228702,22656505,71273364,224965675,712249471,2261326010,
%U A105641 7197988973,22966210236,73437955105,235307698544,755395560220,2429293941019
%N A105641 Number of hill-free Dyck paths of semilength n, having no UUDD's, where U=(1,1) and D=(1,-1) (a hill in a Dyck path is a peak at level 1).
%C A105641 a(n) = A105640(n,0).
%H A105641 E. Deutsch and L. Shapiro, <a href="https://doi.org/10.1016/S0012-365X(01)00121-2">A survey of the Fine numbers</a>, Discrete Math., 241 (2001), 241-265.
%F A105641 G.f.: [(1+z)^2-sqrt((1+z^2)^2-4z)]/[2z(2+z+z^2)]-1.
%F A105641 D-finite with recurrence 2*(n+1)*a(n) +(-7*n+5)*a(n-1) +(n-5)*a(n-2) +2*(-n-1)*a(n-3) +2*(2*n-7)*a(n-4) +(n-5)*a(n-5) +(n-5)*a(n-6)=0. - _R. J. Mathar_, Jul 24 2022
%e A105641 a(4)=2 because we have UUDUDUDD and UUUDUDDD.
%p A105641 G:=((1+z)^2-sqrt((1+z^2)^2-4*z))/2/z/(2+z+z^2)-1: Gser:=series(G,z=0,36): seq(coeff(Gser,z^n),n=2..32);
%Y A105641 Cf. A118995.
%K A105641 nonn
%O A105641 2,3
%A A105641 _Emeric Deutsch_, May 08 2006