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%I #11 Jul 26 2022 12:08:47
%S 1,1,2,4,12,37,119,390,1307,4460,15452,54207,192170,687386,2477810,
%T 8992007,32825653,120460613,444125661,1644324767,6111002752,
%U 22789116600,85251100275,319826371389,1203008722282,4536009027311,17141555233270
%N Number of Dyck paths of semilength n having no UUUDDD's starting at level zero; here U=(1,1), D=(1,-1). Also number of Dyck paths of semilength n having no UUDUDD's starting at level zero.
%C Column 0 of A114499.
%H G. C. Greubel, <a href="/A114500/b114500.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: 1/(1 - z*C + z^3), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan function.
%F a(n) ~ 4^(n+5)/(1089*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 20 2014
%F D-finite with recurrence +(n+1)*a(n) +2*(-2*n+1)*a(n-1) +(n+1)*a(n-2) +2*(-2*n+1)*a(n-3) +(n+1)*a(n-5) +2*(-2*n+1)*a(n-6)=0. - _R. J. Mathar_, Jul 26 2022
%e a(4)=12 because among the 14 Dyck paths of semilength 4 only UDUUUDDD and UUUDDDUD contain UUUDDD starting at level 0.
%p C:=(1-sqrt(1-4*z))/2/z: G:=1/(1-z*C+z^3): Gser:=series(G,z=0,35): 1,seq(coeff(Gser,z^n),n=1..30);
%t CoefficientList[Series[1/(1-x*(1-Sqrt[1-4*x])/2/x+x^3), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o (PARI) x='x+O('x^50); Vec(1/(1-x*(1-sqrt(1-4*x))/(2*x*(1+x^2))) \\ _G. C. Greubel_, Mar 17 2017
%Y Cf. A114499.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Dec 04 2005
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