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A114589 Number of hill-free Dyck paths of semilength n+3 and having no peaks at even levels (a hill in a Dyck path is a peak at level 1).

%I A114589 #19 Sep 09 2024 22:43:30
%S A114589 1,1,3,7,17,43,110,286,753,2003,5376,14540,39589,108427,298512,825664,
%T A114589 2293271,6393539,17885835,50191175,141247519,398537101,1127203038,
%U A114589 3195229662,9076078057,25830193513,73643406563,210312889095
%N A114589 Number of hill-free Dyck paths of semilength n+3 and having no peaks at even levels (a hill in a Dyck path is a peak at level 1).
%C A114589 Column 0 of A114588. The number of hill-free Dyck paths having no peaks at odd level are given by the Riordan numbers (A005043).
%C A114589 From _Paul Barry_, Jul 05 2009: (Start)
%C A114589 The sequence 1,0,0,1,1,3,7,...
%C A114589 has g.f. ((1+x)*(1+2*x)-sqrt((1+x)*(1-3*x)))/(2*x*(2+2*x+x^2)).
%C A114589 It is the inverse binomial transform of A035929(n+1). (End)
%H A114589 G. C. Greubel, <a href="/A114589/b114589.txt">Table of n, a(n) for n = 0..1000</a>
%F A114589 G.f.: (1 -z -2*z^2 -2*z^3 -sqrt(1-3*z^2-2*z))/(2*z^4*(2+2*z+z^2)).
%F A114589 a(n) ~ 3^(n+11/2) / (50*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 20 2014
%F A114589 Conjecture: 2*(n+4)*a(n) +2*(-n-1)*a(n-1) +3*(-3*n-4)*a(n-2) +(-8*n-11)*a(n-3) +3*(-n-1)*a(n-4)=0. - _R. J. Mathar_, Jul 02 2018
%e A114589 a(2)=3 because we have UUUDDUUDDD, UUUDUDUDDD and UUUUUDDDDD, where U=(1,1), D=(1,-1).
%p A114589 G:=(1-z-2*z^2-2*z^3-sqrt(1-3*z^2-2*z))/2/z^4/(2+2*z+z^2): Gser:=series(G,z=0,35): 1, seq(coeff(Gser,z^n),n=1..30);
%t A114589 CoefficientList[Series[(1-x-2*x^2-2*x^3-Sqrt[1-3*x^2-2*x])/2/x^4 /(2+2*x+x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o A114589 (PARI) x='x+('x^50); Vec((1-x-2*x^2-2*x^3-sqrt(1-3*x^2-2*x))/(2*x^4*(2+2*x+x^2))) \\ _G. C. Greubel_, Mar 17 2017
%Y A114589 Cf. A114588, A005043.
%K A114589 nonn
%O A114589 0,3
%A A114589 _Emeric Deutsch_, Dec 11 2005