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A114627 Number of hill-free Dyck paths of semilength n+3 and having no peaks at level 2 (a Dyck path is said to be hill-free if it has no peaks at level 1).

%I A114627 #17 Sep 09 2024 22:42:58
%S A114627 1,2,6,19,61,202,683,2348,8184,28855,102731,368813,1333684,4853436,
%T A114627 17761181,65320691,241300829,894958140,3331323651,12441078958,
%U A114627 46601721324,175040968111,659136721385,2487852579751,9410480922018
%N A114627 Number of hill-free Dyck paths of semilength n+3 and having no peaks at level 2 (a Dyck path is said to be hill-free if it has no peaks at level 1).
%C A114627 Column 0 of A114626.
%H A114627 G. C. Greubel, <a href="/A114627/b114627.txt">Table of n, a(n) for n = 0..1000</a>
%F A114627 G.f.: (C-1)/(z*(1+z+z^2-z*(1+z)*C)), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan function.
%F A114627 a(n) ~ 4^(n+5) / (121*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 20 2014
%F A114627 D-finite with recurrence +(n+3)*a(n) +(-n+3)*a(n-1) +2*(-5*n-6)*a(n-2) +(-7*n-9)*a(n-3) +2*(-2*n-3)*a(n-4)=0. - _R. J. Mathar_, Jul 26 2022
%e A114627 a(1)=2 because we have UUUDUDDD and UUUUDDDD, where U=(1,1), D=(1,-1).
%p A114627 C:=(1-sqrt(1-4*z))/2/z: G:=(C-1)/z/(1+z+z^2-z*(1+z)*C): Gser:=series(G,z=0,32): 1,seq(coeff(Gser,z^n),n=1..28);
%t A114627 CoefficientList[Series[((1-Sqrt[1-4*x])/2/x-1)/x/(1+x+x^2-x*(1+x)*(1-Sqrt[1-4*x])/2/x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o A114627 (PARI) x='x+O('x^50); Vec((1-2*x-sqrt(1-4*x))/(x^2*(2*x^2+x+1+(1+x)*sqrt(1-4*x)))) \\ _G. C. Greubel_, Mar 18 2017
%Y A114627 Cf. A000108, A114626.
%K A114627 nonn
%O A114627 0,2
%A A114627 _Emeric Deutsch_, Dec 18 2005